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This article is cited in 3 scientific papers (total in 3 papers) Left-invariant optimal control problems on Lie groups that are integrable by elliptic functions Yu. L. Sachkov Ailamazyan Program Systems Institute of Russian Academy of Sciences
Bibliographic databases:
     1. PrefaceStudying left-invariant control systems on Lie groups and homogeneous spaces is one of the central topics in geometric control theory. This is a natural and important class of systems from the theoretical standpoint, for which an interesting global theory can be developed (such systems arise, for example, when we consider a local nilpotent approximation to smooth systems). On the other hand, such systems can model a number of applied problems (rotation and rolling of rigid bodies, robot motion, quantum mechanics, computer vision). It is well known that it is very difficult to find an explicit solution of a global non-linear control problem (for example, a problem of controllability or an optimal control problem) when it does not have a large symmetry group. For invariant problems on Lie groups (and their projections onto homogeneous spaces) we can often find an explicit solution by means of the methods of geometric control theory and using the techniques of differential geometry and the theory of Lie groups and algebras. The resulting solution of the invariant problem can be a good approximation to the cor- responding non-linear problem. For instance, the invariant sub-Riemannian geometry of the Heisenberg group is the cornerstone of all sub-Riemannian geometry. The two main problems considered for left-invariant systems are the controllability problem and the optimal control problem. There are many publications concerning the controllability problem, for instance, [128]. In this survey we consider only problems integrable by elliptic functions. Problems integrable by elementary functions were considered in [143]. 2. Problems integrable by elliptic functions and integrals2.1. Elliptic integrals and functionsStandard references on elliptic integrals and functions are [8], [98], and [162]. Below we provide some minimal information about these that is necessary for the next sections. Elliptic integrals in the Jacobi form: Legendre elliptic integrals of the first kind
$$
\begin{equation*}
F(\varphi,k)=\int_0^{\varphi}\frac{dt}{\sqrt{1-k^2\sin^2 t}}\,,
\end{equation*}
\notag
$$
integrals of the second kind
$$
\begin{equation*}
E(\varphi, k)=\int_0^{\varphi} \sqrt{1-k^2 \sin^2 t} \, dt,
\end{equation*}
\notag
$$
integrals of the third kind
$$
\begin{equation*}
\Pi(m; \varphi, k)=\int_0^{\varphi} \frac{dt}{(1+m \sin^2 t)\sqrt{1-k^2 \sin^2 t}}\,;
\end{equation*}
\notag
$$
here and in what follows the elliptic modulus $k$ lies in the interval $(0,1)$. The complementary modulus is $k'=\sqrt{1-k^2}$ . The complete elliptic integrals are
$$
\begin{equation*}
K(k)=F\biggl(\frac{\pi}{2}\,, k\biggr)\quad\text{and} \quad E(k)=E\biggl(\frac{\pi}{2}\,, k\biggr).
\end{equation*}
\notag
$$
Jacobi elliptic functions:
$$
\begin{equation*}
\begin{gathered} \, \varphi=\operatorname{am}(u,k) \ \ \Longleftrightarrow \ \ u=F(\varphi, k), \\ \operatorname{sn}(u,k)=\sin \operatorname{am}(u,k), \qquad \operatorname{cn}(u,k)=\cos \operatorname{am}(u,k), \\ \operatorname{dn}(u,k)=\sqrt{1-k^2 \operatorname{sn}^2(u,k)}\,, \qquad \operatorname{E}(u,k)=E(\operatorname{am}u,k). \end{gathered}
\end{equation*}
\notag
$$
In the notation for elliptic functions the modulus $K$ is often dropped. Standard formulae. Derivatives and integrals:
$$
\begin{equation*}
\operatorname{am}' u=\operatorname{dn} u, \quad \operatorname{sn}'u=\operatorname{cn} u \operatorname{dn} u, \quad \operatorname{cn}' u=- \operatorname{sn} u \operatorname{dn} u, \quad \operatorname{dn}'u=- k^2 \operatorname{sn} u \operatorname{cn} u
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\int_0^u \operatorname{dn}^2 t \, dt=\operatorname{E}(u).
\end{equation*}
\notag
$$
Degenerate cases:
$$
\begin{equation*}
\begin{alignedat}{6} k &\to +0 &\ \ &\Longrightarrow &\ \ \operatorname{sn} u &\to \sin u, &\ \ \operatorname{cn} u &\to \cos u,&\ \ \operatorname{dn} u &\to 1, &\ \ \operatorname{E}(u) &\to u; \\ k &\to 1-0 &\ \ &\Longrightarrow &\ \ \operatorname{sn} u &\to \tanh u, &\ \ \operatorname{cn} u&\to \frac{1}{\operatorname{ch} u}\,, &\ \ \operatorname{dn} u &\to \frac{1}{\cosh u}\,, &\ \ \operatorname{E}(u) &\to \tanh u. \end{alignedat}
\end{equation*}
\notag
$$
2.2. A mathematical pendulumIn all sub-Riemannian problems presented in §§ 2.3–2.10 the vertical subsystem of the Hamiltonian system of the Pontryagin maximum principle reduces mysteriously to the pendulum equation, so that all of them are integrated by elliptic functions and integrals. 2.2.1. The pendulum equation and its solutionConsider a mathematical pendulum, which is a point mass fixed on a weightless inextensible rod of length $L$, which rotates freely in a vertical plane about the point of suspension. Let $\theta$ denote the angle between the pendulum and its downward vertical position. Then the motion of the pendulum satisfies the equations where $r=g/L > 0$ and $g$ is the gravitational acceleration. The total energy of the pendulum (which is a first integral of equations (2.1)) is The motion pattern of the pendulum is determined by the value of $E$:
We have described above the motion pattern of the pendulum (2.1) for $r=g/L >0$. On the other hand, if $r=0$ (which can be interpreted as the absence of gravitational forces), then
The case $r < 0$ (when the gravitational force is upward) reduces to $r > 0$ after the substitution $(\theta, c, r) \mapsto (\theta+\pi, c, -r)$. 2.2.2. Straightening coordinatesFor $r > 0$ the phase cylinder of the pendulum system (2.1),
$$
\begin{equation*}
C=\{(\theta, c) \mid \theta\in S^1, \ c \in \mathbb{R}\}, \qquad S^1=\mathbb{R}/{2\pi}\mathbb{Z},
\end{equation*}
\notag
$$
is stratified depending on the motion pattern: where
$$
\begin{equation*}
\begin{aligned} \, C_1&=\{(\theta, c) \in C \mid E \in (-r, r)\}, \\ C_2&=\{(\theta, c) \in C \mid E >r\}, \\ C_3&=\{(\theta, c) \in C \mid E=r, \ c \ne 0\}, \\ C_4&=\{(\theta, c) \in C \mid c=0, \ \theta=0\}, \\ C_5&=\{(\theta, c) \in C \mid c=0, \ \theta=\pi\}. \end{aligned}
\end{equation*}
\notag
$$
In the domains $C_1$, $C_2$, and $C_3$ we can introduce coordinates $(\varphi,k)$ straightening the equation of the pendulum. If $(\theta,c) \in C_1$, then set
$$
\begin{equation*}
\begin{gathered} \, k=\sqrt{\frac{E+r}{2r}} \in (0, 1), \qquad \sqrt r\,\varphi \in [0,4K(k)] \ \operatorname{mod}{4K(k)}, \\ \sin \frac{\theta}{2}=k \operatorname{sn}(\sqrt r\,\varphi, k), \qquad \cos \frac{\theta}{2}=\operatorname{dn}(\sqrt r\,\varphi, k), \\ c=2 k \sqrt r\,\operatorname{cn}(\sqrt r\,\varphi, k). \end{gathered}
\end{equation*}
\notag
$$
If $(\theta, c) \in C_2$, then set
$$
\begin{equation*}
\begin{gathered} \, k=\sqrt{\frac{2r}{E+r}} \in (0, 1), \qquad \sqrt r\,\varphi\in [0,2k K(k)] \ \operatorname{mod}{2k K(k)}, \\ \sin \frac{\theta}{2}=\pm\operatorname{sn} \biggl(\frac{\sqrt r\,\varphi}{k}\,, k\biggr), \qquad \cos \frac{\theta}{2}=\operatorname{cn} \biggl(\frac{\sqrt r\,\varphi}{k}\,, k\biggr), \\ c=\pm 2 \frac{\sqrt r}{k}\operatorname{dn} \biggl(\frac{\sqrt r \,\varphi}{k}\,,k\biggr), \qquad \pm=\operatorname{sign} c. \end{gathered}
\end{equation*}
\notag
$$
If $(\theta, c) \in C_3$, then set
$$
\begin{equation*}
\begin{gathered} \, k=1, \quad \varphi \in \mathbb{R}, \\ \sin \frac{\theta}{2}=\pm \tanh(\sqrt r\,\varphi), \qquad \cos \frac{\theta}{2}=\frac{1}{\cosh(\sqrt r\,\varphi)}\,, \\ c=\pm \frac{2 \sqrt r}{\cosh(\sqrt r\,\varphi)}\,, \qquad \pm=\operatorname{sign} c. \end{gathered}
\end{equation*}
\notag
$$
In the coordinates $(\varphi,k)$ the pendulum equation (2.1) ‘straightens out’: so its solution is
$$
\begin{equation*}
\varphi_t=\varphi+t, \quad k \equiv \operatorname{const}.
\end{equation*}
\notag
$$
We use these straightening coordinates and modifications of these to parametrize the extremal trajectories in §§ 2.4–2.10. 2.2.3. Bibliographic commentsSubsection 2.2.1 is based on [8], while § 2.2.2 is based on [123] (also see [98], [142], and [162]). 2.3. Martinet flat sub-Riemannian problem2.3.1. The statement of the problemThe Martinet flat sub-Riemannian structure is defined by the metric $ds^2=dx^2+dy^2$ on the Martinet distribution $\Delta=\{dz-(1/2)y^2\,dx= 0\}$ in $M=\mathbb{R}^3_{x, y, z}$. An orthonormal frame can be taken in the form
$$
\begin{equation*}
X_1=\frac{\partial}{\partial x}+\frac{y^2}{2}\, \frac{\partial}{\partial z}\,, \qquad X_2=\frac{\partial}{\partial y}\,.
\end{equation*}
\notag
$$
Let $X_3=\partial/\partial z$; then the Lie algebra generated by the fields $X_1$ and $X_2$ has the multiplication table
$$
\begin{equation*}
[X_1,X_2]=-yX_3, \quad [X_2,[X_1,X_2]]=-X_3,\quad [X_1,[X_1,X_2]]=0, \quad \operatorname{ad} X_3=0,
\end{equation*}
\notag
$$
so that this is the Engel algebra (see § 2.9). The Martinet flat sub-Riemannian structure is not left invariant, but we include it in our survey because it plays a special role in sub-Riemannian geometry:
The optimal control problem for the Martinet flat sub-Riemannian structure has the following form:
$$
\begin{equation*}
\begin{gathered} \, \dot q=u_1X_1+u_2X_2, \quad q=(x, y, z) \in\mathbb{R}^3, \quad u=(u_1, u_2)\in\mathbb{R}^2, \\ q(0)=q_0, \quad q(t_1)=q_1, \\ J=\frac{1}{2}\int_0^{t_1}(u_1^2+u_2^2)\,dt \to \min. \end{gathered}
\end{equation*}
\notag
$$
2.3.2. The Pontryagin maximum principleNormal extremals are trajectories of the Hamiltonian field with Hamiltonian
$$
\begin{equation*}
H=\frac{1}{2}(h_1^2+h_2^2)= \frac{1}{2}\biggl[\biggl(p_x+\frac{y^2}{2}p_z\biggr)^2+p_y^2\biggr],
\end{equation*}
\notag
$$
where $(p_x,p_y,p_z)$ are the canonical coordinates of a covector $\lambda \in T^*M$ and $h_i(\lambda)=\langle \lambda,X_i(q) \rangle$, $i=1,2,3$. The corresponding Hamiltonian system $\dot \lambda=\vec H(\lambda)$ has the form
$$
\begin{equation*}
\begin{alignedat}{2} \dot x&=p_x+\frac{y^2}{2}p_z, &\qquad \dot p_x&=0, \\ \dot y&=p_y, &\qquad \dot p_y&=-\biggl(p_x+\frac{y^2}{2}p_z\biggr)p_z y, \\ \dot z&=\biggl(p_x+\frac{y^2}{2}p_z\biggr)\frac{y^2}{2}\,,&\qquad \dot p_z&=0, \end{alignedat}
\end{equation*}
\notag
$$
that is,
$$
\begin{equation}
\begin{alignedat}{2} &\dot x=h_1, &\qquad \dot h_1&=yh_2h_3, \\ &\dot y=h_2, &\qquad \dot h_2&=-yh_1h_3, \\ &\dot z=\frac{y^2}{2}h_1, &\qquad \dot h_3&=0. \end{alignedat}
\end{equation}
\tag{2.2}
$$
We consider extremals on the level surface $\{H=1/2\}$, on which we introduces the coordinates
$$
\begin{equation*}
h_1=\cos \theta, \quad h_2=\sin \theta, \quad\text{and}\quad h_3=c.
\end{equation*}
\notag
$$
2.3.3. SymmetriesReflections. The sub-Riemannian structure $(\Delta,ds^2)$ is invariant under the group of reflections
$$
\begin{equation*}
\begin{gathered} \, \operatorname{Sym}=\{ \operatorname{Id}, \varepsilon^1, \varepsilon^2, \varepsilon^3 \} \cong \mathbb{Z}_2 \times \mathbb{Z}_2, \\ \begin{alignedat}{2} \varepsilon^1\colon (x, y, z)&\mapsto (x, -y, z), &\qquad (\theta, c)&\mapsto (\pi-\theta, c), \\ \varepsilon^2\colon (x, y, z)&\mapsto (- x, y, -z), &\qquad (\theta, c)&\mapsto (-\theta,-c), \\ \varepsilon^3\colon (x, y, z)&\mapsto (- x, -y, -z), &\qquad (\theta, c)&\mapsto (\theta-\pi,-c). \end{alignedat} \end{gathered}
\end{equation*}
\notag
$$
Dilations. The Hamiltonian system (2.2) is invariant under the one-parameter group of dilations
$$
\begin{equation*}
\begin{aligned} \, (x,y,z) &\mapsto (\delta^{-1}x,\delta^{-1}y,\delta^{-3}z), \\ (h_1,h_2,h_3)&\mapsto (\delta^{-1}h_1,\delta^{-1}h_2,\delta h_3). \end{aligned}
\end{equation*}
\notag
$$
2.3.4. A parametrization of geodesicsWe assume below that $q_0=0$. Proposition 2.2. Geodesics with natural parametrization issuing from the point $q_0=0$ are the curves Let $\operatorname{Exp}$ denote the exponential map
$$
\begin{equation*}
\operatorname{Exp}\colon C \times \mathbb{R}_+ \to M, \quad (\lambda, t)\mapsto q_t=\pi \circ e^{t\vec H}(\lambda),
\end{equation*}
\notag
$$
where 2.3.5. Conjugate timeIf a geodesic is strictly normal and its projection onto the $(x, y)$-plane is a straight line, then this geodesic is optimal and therefore free from conjugate points. In the abnormal case the geodesic is optimal but consists of conjugate points. Let $\lambda=(\theta, c) \in C$, and assume that a geodesic $q_t=\operatorname{Exp}(\lambda, t)$ does not project onto the $(x, y)$-plane as a straight line. Because of the symmetries $\varepsilon^1$ and $\varepsilon^2$, we can assume that $c > 0$ and $\theta \in (-\pi/2,\pi/2)$. Then the first conjugate time is
$$
\begin{equation*}
t_{\rm conj}^1(\lambda)=\min\{t > 0 \mid v^2 c_1(v)+vc_2(v)+c_3(v)=0\},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{aligned} \, c_1(v)&=k'^2\,\frac{\operatorname{cn} v}{\operatorname{dn} v}\,, \\ c_2(v)&=k'^2 \operatorname{sn} v- 2k'^2 \operatorname{E}(v)\frac{\operatorname{cn} v}{\operatorname{dn} v}\,, \\ c_3(v)&=\operatorname{E}^2(v)\frac{\operatorname{cn}v}{\operatorname{dn}v}- \operatorname{E}(v)\operatorname{sn} v \end{aligned}
\end{equation*}
\notag
$$
and $v=t\, \sqrt c$ . Theorem 2.1. Let $q_t=\operatorname{Exp}(\lambda,t)$, $\lambda \in C$, $t > 0$, be a geodesic whose projection onto the $(x,y)$-plane is not a straight line. Then As approximate calculations show, the ratio $t_{\rm conj}^1\, \sqrt{|c|}/(3K)$ is roughly a constant, which is equal to $0.97$. 2.3.6. The cut time and cut locusTheorem 2.2. The geodesics whose projections onto the $(x,y)$-plane are straight lines are length minimizers. Given a geodesic $q_t=\operatorname{Exp}(\lambda,t)$, $\lambda \in C$, $t > 0$, whose projection onto the same plane is not a straight line, the cut time on it is $t_{\rm cut}(\lambda)=2K/\sqrt{|c|}$ and corresponds to its first intersection with the Martinet plane $\{y=0\}$. The cul locus is This set is disjoint from the first caustic.2.3.7. A sphere and a wavefrontDilations take different spheres with centre $q_0=0$ to one another, so we can limit ourselves to considering the unit sphere It is shown in Fig. 1 in the coordinates $(x,y,v)$, $v=z-xy^2/6$.Theorem 2.3. The intersection of the sphere $S$ with the cut locus (see Fig. 2) is the curve $k \mapsto \gamma(k)$ in the Martinet plane $\{y=0\}$ given by the parametric equations where $k\in(0,1)$, plus the curve obtained from $\gamma$ by applying the symmetry $\varepsilon^2|_{\{y=0\}}$: $(x,z)\mapsto(-x,-z)$. As $k \to +0$, the curve $\gamma$ becomes the restriction of the graph of an analytic function to the half-plane $\{z>0\}$.As $k \to 1-0$, $\gamma$ becomes the graph of a non-analytic smooth function where $F$ is the flat functionTheorem 2.4. The intersection of the sphere $S$ with the Martinet plane is not subanalytic, so $S$ is not subanalytic either. Consider the unit-time wavefront from $q_0$:
$$
\begin{equation*}
W=\{q \in M \mid q=\operatorname{Exp}(\lambda,1), \ \lambda \in C\};
\end{equation*}
\notag
$$
other wavefronts from $q_0$ can be obtained from it by dilations. Theorem 2.5. The intersection of the wavefront $W$ with the Martinet plane $\{y= 0\}$ and the half-space $\{z > 0\}$ is a union of curves $\gamma_n$, $n \in \mathbb{N}$, whose closure has two branch points $x=\pm 1$, $z=0$. Each $\gamma_n$ is defined by the parametric equations The intersection of $S$ with the Martinet plane and the half-space $\{z > 0\}$ is the curve given parametrically by $k \mapsto (x(k), z(k))$, $k\! \in \! (0,1)$ (see (2.3) and (2.4)). It extends to $\{z \geqslant 0\}$ by continuity by letting $ k\in [0,1]$. The resulting curve is semianalytic for $k \ne 1$. However, it is not semianalytic at $k=1$, so it is not subanalytic. Theorem 2.6. In a neighbourhood of the point $X=0$, where $X=(x+1)/2$, the intersection of the sphere $S$ with the Martinet plane $\{y=0\}$ and the half-plane $\{z \geqslant 0\}$ is the graph of a function of the form where $X \geqslant 0$ and $F$ is an analytic map from a neighbourhood of $(0,0) \in \mathbb{R}^2$ to $\mathbb{R}$. Hence the intersection of $S$ with the Martinet plane is a curve in the $\exp$-$\log$- category [65], [101]. 2.3.8. Bibliographic commentsThis section is based on [3]. 2.4. Sub-Riemannian problem on the group $\operatorname{SE}(2)$ of Euclidean planar motions2.4.1. The problem statementThe mechanical setting. Consider the problem of optimal motion for a kinematic model of a mobile robot in the plane. The state of the robot is described by its position on the plane $(x,y) \in \mathbb{R}^2$ and the angle $\theta\in 2\pi \mathbb{Z}$ of its orientation relative to the positive direction of the $x$-axis. The robot can move at an arbitrary linear velocity $u_1 \in \mathbb{R}$, rotating at an arbitrary angular velocity $u_2\in\mathbb{R}$ in the process. The problem is to take the robot from an initial state $g_0=(x_0,y_0,\theta_0)$ to a terminal state $g_1=(x_1, y_1, \theta_1)$ along a shortest path in the state space. The length of a path in the state space $\mathbb{R}^2_{x, y}\times S^1_{\theta}$ is measured by the integral $\displaystyle\int_0^{t_1}(\dot x^2+\dot y^2+\alpha^2\dot\theta^2)^{1/2}\,dt$, where $\alpha>0$ is a prescribed positive number determining a balance between the linear and angular velocity. The optimal control problem and its normalization. The above problem about a mobile robot can be formalized as an optimal control problem:
$$
\begin{equation*}
\begin{gathered} \, \dot x=u_1 \cos\theta, \quad \dot y=u_1 \sin \theta, \quad \dot\theta=u_2, \\ g=(x, y, \theta) \in \mathbb{R}^2_{x,y}\times S^1_{\theta}, \quad u=(u_1, u_2)\in\mathbb{R}^2, \\ g(0)=g_0, \quad g(t_1)=g_1, \\ l=\int_0^{t_1}\sqrt{u_1^2+\alpha^2u_2^2}\,\, dt \to \min. \end{gathered}
\end{equation*}
\notag
$$
After scaling in the $(x,y)$-plane,
$$
\begin{equation*}
(x,y,\theta) \mapsto \biggl(\frac{x}{\alpha}\,, \frac{y}{\alpha}\,,\theta\biggr),\quad (u_1, u_2)\mapsto \biggl(\frac{u_1}{\alpha}\,,u_2\biggr),
\end{equation*}
\notag
$$
we can reduce this problem to the case when $\alpha=1$. Using parallel translations and rotating the $(x,y)$-plane we can achieve that $g_0=(0,0,0)$. As a result, we obtain the optimal control problem
$$
\begin{equation}
\dot x=u_1 \cos \theta, \quad \dot y=u_1 \sin \theta, \quad \dot \theta=u_2,
\end{equation}
\tag{2.5}
$$
$$
\begin{equation}
g=(x, y, \theta) \in \mathbb{R}^2_{x, y}\times S^1_{\theta}, \qquad u=(u_1, u_2)\in \mathbb{R}^2,
\end{equation}
\tag{2.6}
$$
$$
\begin{equation}
g(0)=g_0=(0, 0, 0), \quad g(t_1)= g_1=(x_1, y_1, \theta_1),
\end{equation}
\tag{2.7}
$$
This is the sub-Riemannian problem specified by the orthonormal frame
$$
\begin{equation}
X_1=\cos\theta\,\frac{\partial}{\partial x}+ \sin \theta\,\frac{\partial}{\partial y}\,, \quad X_2=\frac{\partial}{\partial \theta}\,.
\end{equation}
\tag{2.9}
$$
The group of plane motions. The group $G=\operatorname{SE}(2)$ of proper Euclidean motions of the plane is a semidirect product of the group of parallel translations $\mathbb{R}^2$ and the group of rotations $\operatorname{SO}(2)$:
$$
\begin{equation*}
\operatorname{SE}(2)=\mathbb{R}^2 \rtimes \operatorname{SO}(2).
\end{equation*}
\notag
$$
It has the linear representation
$$
\begin{equation*}
\operatorname{SE}(2)=\left\{\begin{pmatrix} \cos \theta &-\sin \theta & x \\ \sin \theta &\hphantom{-} \cos \theta & y \\ 0 & 0 & 1 \end{pmatrix}\ \bigg|\ \theta \in S^1= \mathbb{R}/2\pi\mathbb{Z}, \ x,y \in \mathbb{R}\right\}.
\end{equation*}
\notag
$$
We can calculate the action of a motion $g=(x, y, \theta)$ on a vector $(a, b) \in \mathbb{R}^2$ using matrix multiplication:
$$
\begin{equation*}
\begin{pmatrix} \cos \theta & -\sin \theta & x \\ \sin \theta & \hphantom{-}\cos \theta & y \\ 0 & 0 & 1 \end{pmatrix}\cdot \begin{pmatrix} a \\ b \\ 1 \end{pmatrix}=\begin{pmatrix} a \cos \theta-b \sin\theta+x \\ a\sin\theta+b\cos \theta+y \\ 1 \end{pmatrix},
\end{equation*}
\notag
$$
so that
$$
\begin{equation*}
g\colon (a,b) \mapsto (a\cos\theta-b\sin\theta+x, \ a\sin\theta+b \cos\theta+y).
\end{equation*}
\notag
$$
The Lie algebra of the group $\operatorname{SE}(2)$ is
$$
\begin{equation*}
\mathfrak{g}=\mathfrak{se}(2)=\operatorname{span}(E_{21}-E_{12},E_{13},E_{23}),
\end{equation*}
\notag
$$
where $E_{ij}$ is the $ 3\times 3 $ matrix containing a unique non-trivial entry, namely, $1$ in row $i$ and column $j$. Basis left-invariant vector fields on $\operatorname{SE}(2)$ are
$$
\begin{equation*}
\begin{gathered} \, X_1=g E_{13}=\cos\theta\,\frac{\partial}{\partial x}+ \sin \theta\,\frac{\partial}{\partial y}\,, \\ X_2=g(E_{21}-E_{12})=\frac{\partial}{\partial \theta}\,, \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
X_3=- g E_{23}=\sin\theta\,\frac{\partial}{\partial x}- \cos \theta\,\frac{\partial}{\partial y}
\end{equation*}
\notag
$$
and their multiplication table is
$$
\begin{equation}
[X_1,X_2]=X_3, \quad [X_2,X_3]=X_1, \quad [X_1,X_3]=0.
\end{equation}
\tag{2.10}
$$
The orthonormal frame (2.9) for the sub-Riemannian problem (2.5)–(2.8) consists of left-invariant fields, so this is a left-invariant sub-Riemannian problem on the group $G=\operatorname{SE}(2)$. According to the Agrachev–Barilari classification [1], this is the unique, up to local isometries, totally non-holonomic sub-Riemannian problem on $\operatorname{SE}(2)$; the corresponding invariants are $\chi=\kappa=1$. That optimal controls exist in (2.5)–(2.8) is a consequence of the Rashevskii–Chow and Filippov theorems: the system has full rank because
$$
\begin{equation*}
\mathfrak{g}=\operatorname{span}(X_1,X_2,X_3), \qquad X_3=[X_1,X_2].
\end{equation*}
\notag
$$
2.4.2. The Pontryagin maximum principleAbnormal trajectories are constant in time. Normal extremals are trajectories of the Hamiltonian system $\dot\lambda=\vec H(\lambda)$, $\lambda \in T^*G$, where $H=(h_1^2+h_2^2)/2$, $h_i(\lambda)=\langle \lambda,X_i \rangle$, $i=1,2,3$. We can write this system in coordinates as follows:
$$
\begin{equation}
\begin{alignedat}{3} \dot h_1&=- h_2 h_3, &\quad \dot h_2&=h_1 h_3, &\quad \dot h_3&=h_1 h_2, \\ \dot x&=h_1 \cos \theta, &\quad \dot y&=h_1 \sin \theta, &\quad \dot \theta&=h_2. \nonumber \end{alignedat}
\end{equation}
\tag{2.11}
$$
On the level surface $\{H=1/2\}$, in the coordinates $(\gamma, c)$, where
$$
\begin{equation*}
h_1=\sin \frac{\gamma}{2}\,, \quad h_2=-\cos \frac{\gamma}{2}\,, \quad\text{and}\quad c=2h_3,
\end{equation*}
\notag
$$
the vertical subsystem (2.11) of this Hamiltonian system takes the form of a two- sheeted cover of the pendulum system:
$$
\begin{equation}
\dot\gamma=c, \quad \dot c=-\sin\gamma, \quad (\gamma,c) \in C=\mathfrak{g}^*\cap\biggl\{H=\frac{1}{2}\biggr\} \cong (2S^1_{\gamma})\times \mathbb{R}_c, \quad 2S^1=\mathbb{R}/4\pi\mathbb{Z}.
\end{equation}
\tag{2.12}
$$
A first integral of this equation is equal to the energy of the pendulum The symplectic foliation. The Lie coalgebra $\mathfrak{g}^*$ possesses a Casimir function $F=h_1^2+h_3^2$. The symplectic foliation consists of the circular cylinders $\{h_1^2+h_3^2=\operatorname{const} > 0\}$ and the points $\{h_1=h_3=0, \ h_2=\operatorname{const}\}$. The energy of the pendulum is a linear combination of the Casimir function and the Hamiltonian: Stratification of the cylinder $C$ and straightening coordinates. The cylinder $C$ is subdivided into invariant sets of the pendulum (2.12) by critical level curves of the energy $E$:
$$
\begin{equation}
\begin{aligned} \, C&=\bigsqcup_{i=1}^5 C_i, \\ C_1&=\{\lambda \in C \mid E \in (-1, 1) \}, \nonumber \\ C_2&=\{\lambda \in C \mid E \in (1,+\infty) \}, \nonumber \\ C_3&=\{\lambda \in C \mid E =1, \ c \ne 0 \}, \nonumber \\ C_4&=\{\lambda \in C \mid E=- 1 \}= \{ (\gamma, c) \in C \mid \gamma=2 \pi n, \ c=0 \}, \nonumber \\ C_5&=\{\lambda \in C \mid E=1, \ c=0 \}= \{ (\gamma, c) \in C \mid \gamma=\pi+2 \pi n, \ c=0 \}, \nonumber \end{aligned}
\end{equation}
\tag{2.14}
$$
where $n \in \mathbb{Z}$. For the regular integration of the pendulum equation (2.12), on the strata $C_1$, $C_2$, and $C_3$ we introduce variables $(\varphi, k)$ which straighten this equation. If $\lambda=(\gamma, c) \in C_1$, then
$$
\begin{equation*}
\begin{gathered} \, k=\sqrt{\frac{E+1}{2}}= \sqrt{\sin^2 \frac{\gamma}{2}+\frac{c^2}{4}} \in (0,1), \\ \sin \frac{\gamma}{2}=s_1 k \operatorname{sn}(\varphi,k), \qquad s_1=\operatorname{sign} \cos\frac{\gamma}{2}\,, \\ \cos \frac{\gamma}{2}=s_1 \operatorname{dn}(\varphi,k), \\ \frac{c}{2}=k \operatorname{cn}(\varphi,k), \qquad \varphi \in [0, 4 K(k)]. \end{gathered}
\end{equation*}
\notag
$$
If $\lambda=(\gamma, c) \in C_2$, then
$$
\begin{equation*}
\begin{gathered} \, k=\sqrt{\frac{2}{E+1}}=\frac{1}{\sqrt{\sin^2(\gamma/2)+c^2/4}} \in (0,1), \\ \sin \frac{\gamma}{2}= s_2 \operatorname{sn}\biggl(\frac{\varphi}{k}\,,k\biggr), \qquad s_2=\operatorname{sign} c, \\ \cos \frac{\gamma}{2}=\operatorname{cn}\biggl(\frac{\varphi}{k}\,,k\biggr), \\ \frac{c}{2}= \frac{s_2}{k}\operatorname{dn}\biggl(\frac{\varphi}{k}\,,k\biggr), \qquad \varphi \in [0,4 k K(k)]. \end{gathered}
\end{equation*}
\notag
$$
If $\lambda=(\gamma,c) \in C_3$, then
$$
\begin{equation*}
\begin{gathered} \, k=1, \\ \sin \frac{\gamma}{2}=s_1 s_2 \tanh \varphi, \qquad s_1=\operatorname{sign} \cos\frac{\gamma}{2}\,, \quad s_2=\operatorname{sign} c, \\ \cos \frac{\gamma}{2}=\frac{s_1}{\cosh \varphi}\,, \\ \frac{c}{2}=\frac{s_2}{\cosh \varphi}\,, \qquad \varphi \in (-\infty, +\infty). \end{gathered}
\end{equation*}
\notag
$$
The pendulum flow (2.12) straightens out in the variables $(\varphi,k)$:
$$
\begin{equation*}
\dot \varphi=1, \quad \dot k=0, \qquad \lambda=(\varphi,k) \in \bigcup_{i=1}^3 C_i.
\end{equation*}
\notag
$$
A parametrization on geodesics. If $\lambda=(\varphi,k) \in C_1$, then $\varphi_t=\varphi+t$ and
$$
\begin{equation*}
\begin{gathered} \, \begin{aligned} \, \cos \theta_t&=\operatorname{cn} \varphi \operatorname{cn} \varphi_t+ \operatorname{sn} \varphi \operatorname{sn} \varphi_t, \\ \sin \theta_t&=s_1(\operatorname{sn} \varphi \operatorname{cn} \varphi_t- \operatorname{cn} \varphi \operatorname{sn} \varphi_t),\\ \theta_t&=s_1(\operatorname{am} \varphi- \operatorname{am} \varphi_t)\operatorname{mod}{2\pi}, \end{aligned} \\ \begin{aligned} \, x_t&=\frac{s_1}{k}[\operatorname{cn}\varphi(\operatorname{dn}\varphi- \operatorname{dn}\varphi_t)+\operatorname{sn} \varphi(t+\operatorname{E}(\varphi)-\operatorname{E}(\varphi_t))], \\ y_t&=\frac{1}{k}[\operatorname{sn}\varphi(\operatorname{dn}\varphi- \operatorname{dn}\varphi_t)-\operatorname{cn}\varphi(t+ \operatorname{E}(\varphi)-\operatorname{E}(\varphi_t))]. \end{aligned} \end{gathered}
\end{equation*}
\notag
$$
If $\lambda \in C_2$, then
$$
\begin{equation*}
\begin{gathered} \, \begin{aligned} \, \cos \theta_t&=k^2 \operatorname{sn} \psi \operatorname{sn} \psi_t+ \operatorname{dn} \psi \operatorname{dn} \psi_t, \\ \sin \theta_t&=k(\operatorname{sn} \psi \operatorname{dn} \psi_t- \operatorname{dn} \psi \operatorname{sn} \psi_t), \end{aligned} \\ \begin{aligned} \, x_t&=s_2 k\biggl[\operatorname{dn} \psi(\operatorname{cn} \psi- \operatorname{cn} \psi_t)+\operatorname{sn} \psi\,\biggl(\frac{t}{k}+ \operatorname{E}(\psi)-\operatorname{E}(\psi_t)\biggr)\biggr], \\ y_t&=s_2\biggl[k^2\operatorname{sn} \psi (\operatorname{cn} \psi- \operatorname{cn} \psi_t)-\operatorname{dn} \psi\,\biggl(\frac{t}{k}+ \operatorname{E}(\psi)-\operatorname{E}(\psi_t)\biggr)\biggr], \end{aligned} \end{gathered}
\end{equation*}
\notag
$$
where $\psi=\varphi/k$ and $\psi_t=\varphi_t/k=\psi+t/k$. If $\lambda=(\varphi,k) \in C_3$ for $k=1$, then $\varphi_t=\varphi+t$, and
$$
\begin{equation*}
\begin{gathered} \, \begin{aligned} \, \cos \theta_t&=\frac{1}{\cosh \varphi\cosh \varphi_t}+ \tanh \varphi\tanh \varphi_t, \\ \sin \theta_t&= s_1\biggl(\frac{\tanh \varphi}{\cosh \varphi_t}- \frac{\tanh \varphi_t}{\cosh \varphi}\biggr), \end{aligned} \\ \begin{aligned} \, x_t&=s_1 s_2\biggl[\frac{1}{\cosh \varphi} \biggl(\frac{1}{\cosh \varphi}- \frac{1}{\cosh \varphi_t}\biggr)+\tanh \varphi\, (t+\tanh \varphi-\tanh \varphi_t)\biggr], \\ y_t&=s_2\biggl[\tanh \varphi\, \biggl(\frac{1}{\cosh \varphi}- \frac{1}{\cosh \varphi_t}\biggr)-\frac{1}{\cosh \varphi} (t+\tanh \varphi-\tanh \varphi_t)\biggr]. \end{aligned} \end{gathered}
\end{equation*}
\notag
$$
If $\lambda \in C_4$, then If $\lambda \in C_5$, then
$$
\begin{equation*}
\theta_t=0, \qquad x_t=t\operatorname{sign}\sin\frac{\gamma}{2}\,, \qquad y_t=0.
\end{equation*}
\notag
$$
We show the projections of geodesics onto the $(x,y)$-plane in cases $C_1$, $C_2$, and $C_3$ in Figs. 3, 4, and 5, respectively. 2.4.3. Symmetries and Maxwell strataThe phase portrait of the pendulum system (2.12) is invariant under the symmetry group $\operatorname{Sym}$ generated by the reflections of the cylinder $C$ in the $\gamma$- and $c$-coordinate axes and in the origin $(\gamma,c)=(0,0)$ and the rotation through $2\pi$:
$$
\begin{equation*}
\operatorname{Sym}=\{\operatorname{Id},\varepsilon^1,\dots,\varepsilon^7\} \cong \mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{aligned} \, \varepsilon^1\colon (\gamma,c)&\to (\gamma,-c), \\ \varepsilon^2\colon (\gamma,c)&\to (-\gamma,c), \\ \varepsilon^3\colon (\gamma,c)&\to (-\gamma,-c), \\ \varepsilon^4\colon (\gamma,c)&\to (\gamma+2 \pi,c), \\ \varepsilon^5\colon (\gamma,c)&\to (\gamma+2 \pi,-c), \\ \varepsilon^6\colon (\gamma,c)&\to (-\gamma+2 \pi,c), \\ \varepsilon^7\colon (\gamma,c)&\to (-\gamma+2 \pi,-c). \end{aligned}
\end{equation*}
\notag
$$
These symmetries extend in the natural way to the source space and target space of the exponential map. If $\nu=(\lambda,t)=(\gamma,c,t) \!\in\! N=C \times \mathbb{R}_+$, then $\varepsilon^i(\nu)=\nu^i=(\lambda^i,t)=(\gamma^i,c^i,t)\! \in\! N$ and
$$
\begin{equation*}
\begin{aligned} \, (\gamma^1,c^1)&=(\gamma_t, -c_t), \\ (\gamma^2,c^2)&=(-\gamma_t, c_t), \\ (\gamma^3,c^3)&=(-\gamma, -c), \\ (\gamma^4,c^4)&=(\gamma+2 \pi, c), \\ (\gamma^5,c^5)&=(\gamma_t+2 \pi, -c_t), \\ (\gamma^6,c^6)&=(-\gamma_t+ 2 \pi, c_t), \\ (\gamma^7,c^7)&=(-\gamma, -c). \end{aligned}
\end{equation*}
\notag
$$
If $g=(x,y,\theta) \in G$, then $g^i=\varepsilon^i(g)=(x^i,y^i,\theta^i) \in G$, where
$$
\begin{equation*}
\begin{aligned} \, (x^1,y^1,\theta^1)&=(x \cos \theta+y \sin \theta, x \sin \theta- y \cos \theta, \theta), \\ (x^2,y^2,\theta^2)&=(-x \cos \theta- y \sin \theta, -x \sin \theta+y \cos \theta, \theta), \\ (x^3,y^3,\theta^3)&=(-x, -y, \theta), \\ (x^4,y^4,\theta^4)&=(-x, y, -\theta), \\ (x^5,y^5,\theta^5)&=(-x \cos \theta-y \sin \theta, x \sin \theta- y \cos \theta, -\theta), \\ (x^6,y^6,\theta^6)&=(x \cos \theta+y \sin \theta, -x \sin \theta+ y \cos \theta, -\theta), \\ (x^7,y^7,\theta^7)&=(x, -y, -\theta). \end{aligned}
\end{equation*}
\notag
$$
Proposition 2.3. The group $\operatorname{Sym}= \{\operatorname{Id},\varepsilon^1,\dots,\varepsilon^7\}$ is a subgroup of the symmetry group of the exponential map. Theorem 2.7. For almost all geodesics the first Maxwell time corresponding to the symmetry group $\operatorname{Sym}$ has the following expression: Remark 2.1. If for a geodesic the first Maxwell time corresponding to the group $\operatorname{Sym}$ is distinct from $t_{\rm Max}^1$, then it is larger than this quantity, while $t_{\rm Max}^1$ is the first conjugate time. Let $\vec H_v$ denote the vertical component of the Hamiltonian vector field $\vec H$ corresponding to the ordinary differential equation (2.11). Theorem 2.8. The function $t_{\rm Max}^1\colon C \to (0,+\infty]$ has the following invariance properties: 2.4.4. Estimates for the conjugate timeTheorem 2.9. (1) If $\lambda \in C_1\cup C_3\cup C_4\cup C_5$, then $t_{\rm conj}^1(\lambda)=+\infty$. (2) If $\lambda \in C_2$, then $t_{\rm conj}^1(\lambda)\in [2kp_1^1,4kK]$. (3) Therefore, $t_{\rm conj}^1(\lambda) \geqslant t_{\rm Max}^1(\lambda)$ for all $\lambda \in C$. 2.4.5. The diffeomorphism structure of the exponential mapConsider a subset of the state space that contains no fixed points of the maps $\varepsilon^i$:
$$
\begin{equation*}
\begin{gathered} \, \widetilde G=\{g \in G \mid \varepsilon^i(g) \ne g, \ i=1,\dots,7\}= \{g \in G \mid R_1(g) R_2(g)\sin\theta\ne 0\}, \\ \text{where } R_1=y\cos \frac{\theta}{2}-x \sin\frac{\theta}{2} \quad\text{and}\quad R_2=x \cos \frac{\theta}{2}+y\sin\frac{\theta}{2}\,, \end{gathered}
\end{equation*}
\notag
$$
and consider its partition into connected components: where each set $G_i$ is characterized by constant signs of the functions $\sin\theta$, $R_1$, and $R_2$, as indicated in Table 1. Table 1.The definition of the domains $G_i$
Also consider an open dense subset of the space of potentially optimal geodesics:
$$
\begin{equation*}
\widetilde N=\bigl\{(\lambda,t) \in N \mid t < t_{\rm Max}^1(\lambda), \ c_{t/2}\sin\gamma_{t/2} \ne 0\bigr\},
\end{equation*}
\notag
$$
and consider its connected components
$$
\begin{equation*}
\begin{gathered} \, D_1=\bigl\{ (\lambda, t) \in N \mid t < t_{\rm Max}^1(\lambda), \ c_{t/2} > 0, \ \gamma_{t/2}\in(-\pi, 0)\bigr\}, \\ D_2=\bigl\{ (\lambda, t) \in N \mid t < t_{\rm Max}^1(\lambda), \ c_{t/2} > 0, \ \gamma_{t/2}\in(0, \pi)\bigr\}, \\ D_3=\bigl\{ (\lambda, t) \in N \mid t < t_{\rm Max}^1(\lambda), \ c_{t/2} < 0, \ \gamma_{t/2}\in(0, \pi)\bigr\}, \\ D_4=\bigl\{ (\lambda, t) \in N \mid t < t_{\rm Max}^1(\lambda), \ c_{t/2} < 0, \ \gamma_{t/2}\in(-\pi, 0)\bigr\}, \\ D_5=\bigl\{ (\lambda, t) \in N \mid t < t_{\rm Max}^1(\lambda), \ c_{t/2} > 0, \ \gamma_{t/2}\in(\pi, 2\pi)\bigr\}, \\ D_6=\bigl\{ (\lambda, t) \in N \mid t < t_{\rm Max}^1(\lambda), \ c_{t/2} > 0, \ \gamma_{t/2}\in(2\pi, 3\pi)\bigr\}, \\ D_7=\bigl\{ (\lambda, t) \in N \mid t < t_{\rm Max}^1(\lambda), \ c_{t/2} < 0, \ \gamma_{t/2}\in(2\pi, 3\pi)\bigr\}, \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
D_8=\bigl\{ (\lambda, t) \in N \mid t < t_{\rm Max}^1(\lambda), \ c_{t/2} < 0, \ \gamma_{t/2}\in(\pi, 2\pi)\bigr\};
\end{equation*}
\notag
$$
$\widetilde N=\bigsqcup_{i=1}^8D_i$. 2.4.6. The cut timeThe cut time is invariant under the vertical component of the Hamiltonian field $\vec H_v$, so the sub-Riemannian structure on $\operatorname{SE}(2)$ is equioptimal. 2.4.7. The cut locus and its stratification Theorem 2.12. The cut locus is a stratified two-dimensional manifold with the stratification We show the cut locus $\operatorname{Cut} \subset \operatorname{SE}(2)$ in Fig. 6 (in the straightening coordinates $R_1=y \cos(\theta/2)-x \sin(\theta/2)$, $R_2=x \cos(\theta/2)+y \sin(\theta/2)$) and Fig. 7 (as embedded in the solid torus modelling the group $\operatorname{SE}(2)$). 2.4.8. SpheresThe sub-Riemannian spheres $S_R$ are homeomorphic (but not diffeomorphic) to
In Figures 8, 9, and 10 we show the sub-Riemannian spheres of radii $\pi/2$, $\pi$, and $3\pi/2$, respectively, as embedded in the solid torus modelling the group $\operatorname{SE}(2)$. 2.4.9. Metric straight linesThe metric straight lines through the identity element $g_0=\operatorname{Id}$ are $g(t)=\operatorname{Exp}(\lambda,t)$, $t \in \mathbb{R}$, where $\lambda \in C_3\cup C_5$. The geodesics $\operatorname{Exp}(\lambda,t)$, $\lambda \in C_3$, are projected onto tractrices on the $(x,y)$-plane, while geodesics $\operatorname{Exp}(\lambda,t)$, $\lambda \in C_5$, are projected onto the lines $(x,y)=(\pm t,0)$. 2.4.10. The bicycle modelWe can regard the sub-Riemannian problem on $\operatorname{SE}(2)$ as the problem of the optimal motion of a bicycle model. Assume that the front and rear wheels of a bicycle touch the ground at points $\mathbf{f}$ and $\mathbf{b}$, and the distance between these points (wheelbase) is fixed and equal to $\ell$. In the process of motion of the bicycle the points $\mathbf{f}$ and $\mathbf{b}$ draw two curves, the front and rear paths. At each moment of time the line segment $\mathbf{f}-\mathbf{b}$ is tangent to the rear path. We say that the motion is optimal if it minimizes the length of the front path. Then the problem of the optimal motion of a bicycle is just the sub-Riemannian problem (2.5)–(2.8) on the group $\operatorname{SE}(2)$. We say that two curves in the plane have the same shape if one of them can be taken to the other by a composition of motions and dilations. The width of a plane curve is the greatest lower bound of the distances between two parallel straight lines bounding a strip which contains this curve. Theorem 2.13. An optimal trajectory of the front wheel of the bicycle $\mathbf{b}(t)$ is a straight line or an arc of a non-inflectional elastica of width at most $2\ell$. A non- inflectional elastica of any shape can be obtained in this way. Theorem 2.14. An infinite motion of a bicycle is optimal on each interval of motion if and only if it belongs to one of the following two types: 2.4.11. The isometry group and homogeneous geodesicsTheorem 2.15. The isometry group of the sub-Riemannian structure on $\operatorname{SE}(2)$ is where the first factor $\operatorname{SE}(2)$ acts by left shifts on itself, the second factor $\mathbb{Z}_2$ acts on a pair $(\mathbf{b},\mathbf{f})$ as the reflection of the plane in one of the axes, and the third factor $\mathbb{Z}_2$ acts as the reflection $(\mathbf{b},\mathbf{f}) \mapsto (\mathbf{b},2\mathbf{b}-\mathbf{f})$. A geodesic $\gamma$ on a sub-Riemannian manifold $M$ is said to be homogeneous if it is a homogeneous space of a one-parameter subgroup of $\operatorname{Isom}(M)$, that is, if there exists a one-parameter subgroup $\{\varphi_s \mid s \in \mathbb{R}\} \subset \operatorname{Isom}(M)$ such that
A sub-Riemannian manifold is said to be geodesically orbital if all geodesics on it are homogeneous. Theorem 2.16. Homogeneous geodesics on $\operatorname{SE}(2)$ have the form $g(t)=\operatorname{Exp}(\lambda,t)$, $\lambda \in C_4\cup C_5$. They are one-parameter subgroups $e^{t X_2}$ and $e^{t X_1}$; their projections onto the $(x,y)$-plane are the point $(0,0)$ and the line $y=0$. Thus, $\operatorname{SE}(2)$ is not a geodesically orbital space. 2.4.12. Bibliographic commentsSubsections 2.4.1–2.4.3 are based on [116], §§ 2.4.4–2.4.6, 2.4.8, and 2.4.9 on [134], § 2.4.7 on [135], and §§ 2.4.10 and 2.4.11 on [12] and [140]. The sub-Riemannian problem on $\operatorname{SE}(2)$ was also considered in [2], [12], [39], [111], and [142]. 2.5. The sub-Riemannian problem on the motion group $\operatorname{SH}(2)$ of the pseudo-Euclidean plane2.5.1. The motion group $\operatorname{SH}(2)$ of the pseudo- Euclidean planeThe pseudo- Euclidean plane is a two-dimensional real linear space endowed with the sign- indefinite bilinear form
$$
\begin{equation*}
(\mathbf{x},\mathbf{y})=x_1 y_1-x_2 y_2, \qquad \mathbf{x}=(x_1, x_2), \quad \mathbf{y}=(y_1, y_2).
\end{equation*}
\notag
$$
The distance $r$ between two points $\mathbf{x}=(x_1,x_2)$ and $\mathbf{y}=(y_1,y_2)$ is defined by
$$
\begin{equation*}
r^2=(\mathbf{x}-\mathbf{y},\mathbf{x}-\mathbf{y})= (x_1-y_1)^2-(x_2-y_2)^2,\qquad r=\begin{cases} |r| &\text{ for }\ r^2 \geqslant 0, \\ i |r| &\text{ for }\ r^2 < 0. \end{cases}
\end{equation*}
\notag
$$
The locus of points $\mathbf{x}=(x_1,x_2)$ lying at distance zero from the origin $(x_1^2-x_2^2=0$) is called the light cone. The complement of the pseudo-Euclidean plane to the light cone falls into four connected components, quadrants ($\operatorname{sign}(x_1-x_2)=\pm 1$, $\operatorname{sign}(x_1+x_2 )=\pm 1$). The Lie group $\operatorname{SH}(2)$ and the Lie algebra $\mathfrak{sh}(2)$. A motion of the pseudo- Euclidean plane is a linear transformation that preserves the orientation, the quadrants, and the distances between points in the plane. The group of motions of the pseudo-Euclidean plane is denoted by $\operatorname{SH}(2)$. It has the linear representation
$$
\begin{equation*}
\operatorname{SH}(2)=\left\{\begin{pmatrix} \cosh z & \sinh z & x \\ \sinh z & \cosh z & y \\ 0 & 0 & 1 \end{pmatrix}\ \bigg|\ x, y, z \in \mathbb{R}\right\}.
\end{equation*}
\notag
$$
The action of a motion $g=(x,y,z)$ on a point $\mathbf{a}=(a_1,a_2)$ in the pseudo-Euclidean plane can be calculated using matrix multiplication:
$$
\begin{equation*}
\begin{pmatrix} \cosh z & \sinh z & x \\ \sinh z & \cosh z & y \\ 0 & 0 & 1 \end{pmatrix}\begin{pmatrix} a_1 \\ a_2 \\ 1 \end{pmatrix}=\begin{pmatrix} a_1 \cosh z+a_2 \sinh z+x \\ a_1 \sinh z+a_2 \cosh z+y \\ 1 \end{pmatrix},
\end{equation*}
\notag
$$
so that $g \colon (a_1,a_2) \mapsto (a_1 \cosh z+a_2 \sinh z+x, a_1 \sinh z+a_2 \cosh z+y)$. $G=\operatorname{SH}(2)$ is a Lie group with Lie algebra
$$
\begin{equation*}
\mathfrak{g}=\mathfrak{sh}(2)= \operatorname{span}(E_{21}+E_{12},E_{13},E_{23}).
\end{equation*}
\notag
$$
Basis left-invariant vector fields on $\operatorname{SH}(2)$ are
$$
\begin{equation*}
\begin{gathered} \, X_1=L_{g*} E_{13}=\cosh z\,\frac{\partial}{\partial x}+ \sinh z\,\frac{\partial}{\partial y}\,, \\ X_2=L_{g*} (E_{21}+E_{12})=\frac{\partial}{\partial z}\,, \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
X_3=L_{g*} E_{23}=\sinh z\,\frac{\partial}{\partial x}+ \cosh z\,\frac{\partial}{\partial y}
\end{equation*}
\notag
$$
with the multiplication table
$$
\begin{equation}
[X_1, X_2]=- X_3, \quad [X_2. X_3]=X_1, \quad [X_1, X_3]=0.
\end{equation}
\tag{2.15}
$$
2.5.2. The sub-Riemannian problem on $\operatorname{SH}(2)$Consider the sub-Riemannian problem on the group $\operatorname{SH}(2)$ with orthonormal frame $(X_1, X_2)$:
$$
\begin{equation}
\dot g=u_1 X_1+u_2 X_2, \quad g \in G=\operatorname{SH}(2), \quad u=(u_1, u_2) \in \mathbb{R}^2,
\end{equation}
\tag{2.16}
$$
By the Agrachev–Barilari classification [1] this is the unique, up to local isometries, non-integrable sub-Riemannian problem of rank 2 on $\operatorname{SH}(2)$; the corresponding invariants are $\chi=-\kappa=1$. 2.5.3. GeodesicsThat optimal controls exist in problem (2.16)–(2.18) follows from the Rashevskii–Chow and Filippov theorems. The Pontryagin maximum principle. Abnormal trajectories are constant in time. Normal extremals are the projections of trajectories of the Hamiltonian system $\dot\lambda=\vec{H}(\lambda)$, $\lambda \in T^*G$, where $H=(h_1^2+h_2^2)/2$, $h_i(\lambda)=\langle \lambda,X_i\rangle$, $i=1,2,3$. In coordinates this system looks as follows: On the level surface $\{H=1/2\}$, in the coordinates $(\gamma,c)$, where
$$
\begin{equation*}
h_1=\cos \frac{\gamma}{2}\,, \quad h_2=\sin \frac{\gamma}{2}\,, \quad c=-2 h_3,
\end{equation*}
\notag
$$
the vertical subsystem (2.19)–(2.21) assumes the form of a two-sheeted cover of the pendulum system:
$$
\begin{equation}
\dot\gamma=c, \quad \dot c=-\sin \gamma, \qquad (\gamma,c) \in \mathfrak{g}^* \cap \biggl\{H=\frac{1}{2}\biggr\} \simeq (2 S^1_{\gamma}) \times \mathbb{R}_c.
\end{equation}
\tag{2.22}
$$
A first integral of this equation is the energy of the pendulum
$$
\begin{equation*}
E=\frac{c^2}{2}-\cos \gamma=2 h_3^2-h_1^2+h_2^2 \in [-1,+\infty).
\end{equation*}
\notag
$$
The symplectic foliation. The Lie coalgebra $\mathfrak{g}^*$ possesses a Casimir function $F=h_1^2-h_3^2$. The symplectic foliation consists of
The stratification of the cylinder $C$ and straightening coordinates. Since the vertical subsystem of the Hamiltonian system for the problem on $\operatorname{SH}(2)$, the pendulum system (2.22), coincides with a similar system (2.12) for the problem on $\operatorname{SE}(2)$, the stratification of $C$ and straightening coordinates $(\varphi,k)$ for the problem on $\operatorname{SH}(2)$ also coincide with the ones for the problem on $\operatorname{SE}(2)$ (see § 2.4.2). A parametrization of geodesics. If $\lambda=(\varphi,k) \in C_1$, then $\varphi_t=\varphi+t$ and
$$
\begin{equation*}
\begin{pmatrix} x\\ y\\ z \end{pmatrix}={\begin{pmatrix} \dfrac{s_{1}}{2}\biggl[\biggl(w+\dfrac{1}{w(1-k^{2})}\biggr) [\operatorname{E}(\varphi)-\operatorname{E}(\varphi_{0})]+ \biggl(\dfrac{k}{w(1-k^{2})}-kw\biggr)[\operatorname{sn}\varphi- \operatorname{sn}\varphi_{0}]\biggr] \\ \dfrac{1}{2}\biggl[\biggl(w-\dfrac{1}{w(1-k^{2})}\biggr) [\operatorname{E}(\varphi)-\operatorname{E}(\varphi_{0})]- \biggl(\dfrac{k}{w(1-k^{2})}+kw\biggr)[\operatorname{sn}\varphi- \operatorname{sn}\varphi_{0}]\biggr] \\ s_{1}\log[(\operatorname{dn}\varphi-k\operatorname{cn}\varphi)w] \end{pmatrix}},
\end{equation*}
\notag
$$
where $w=1/(\operatorname{dn}\varphi_{0}-k\operatorname{cn}\varphi_{0})$. If $\lambda=(\varphi,k) \in C_2$, then $\psi=\varphi/k$, $\psi_t=\varphi_t/k=\psi+t/k$ and
$$
\begin{equation*}
\begin{aligned} \, x&=\frac{1}{2}\biggl(\frac{1}{w(1-k^{2})}-w\biggr)[\operatorname{E}(\psi)- \operatorname{E}(\psi_{0})-k^{\prime2}(\psi-\psi_{0})] \\ &\qquad+\frac{1}{2}(kw+\frac{k}{w(1-k^{2})}) [\operatorname{sn}\psi-\operatorname{sn}\psi_{0}], \\ y&=-\frac{s_{2}}{2}\biggl(\frac{1}{w(1-k^{2})}+w\biggr) [\operatorname{E}(\psi)-\operatorname{E}(\psi_0)-k^{\prime2}(\psi-\psi_{0})] \\ &\qquad+\frac{s_{2}}{2}\biggl(kw-\frac{k}{w(1-k^{2})}\biggr) [\operatorname{sn}\psi-\operatorname{sn}\psi_{0}], \\ z&=s_{2}\log[(\operatorname{dn}\psi-k\operatorname{cn}\psi)w], \end{aligned}
\end{equation*}
\notag
$$
where $w=1/(\operatorname{dn}\psi_{0}-k\operatorname{cn}\psi_{0})$. If $\lambda=(\varphi,k) \in C_3$, $k=1$, then ${\varphi_t}=\varphi+t$ and
$$
\begin{equation*}
\begin{pmatrix} x\\ y\\ z \end{pmatrix}=\begin{pmatrix} \dfrac{s_{1}}{2}\biggl[\dfrac{1}{w}(\varphi-\varphi_{0})+ w(\tanh \varphi-\tanh \varphi_{0})\biggr] \\ \dfrac{s_{2}}{2}\biggl[\dfrac{1}{w}(\varphi-\varphi_{0})- w(\tanh \varphi-\tanh \varphi_{0})\biggr] \\ -s_{1}s_{2}\log(w\operatorname{sech}\varphi) \end{pmatrix},
\end{equation*}
\notag
$$
where $w=\cos h \varphi_{0}$. If $\lambda=(\gamma,c) \in C_4$, then
$$
\begin{equation*}
\begin{pmatrix} x\\ y\\ z \end{pmatrix}=\begin{pmatrix} \operatorname{sign}\biggl(\cos\dfrac{\gamma}{2}\biggr)t \\ 0 \\ 0 \end{pmatrix}.
\end{equation*}
\notag
$$
If $\lambda=(\gamma,c) \in C_5$, then
$$
\begin{equation*}
\begin{pmatrix} x\\ y\\ z \end{pmatrix}=\begin{pmatrix} 0\\ 0\\ \operatorname{sign}\biggl(\sin\dfrac{\gamma}{2}\biggr)t \end{pmatrix}.
\end{equation*}
\notag
$$
The projection of the geodesic onto the $(x,y)$-plane has curvature $\tan(\gamma/2)/\cosh^{3/2}(2z)$. It has inflection points for $\sin(\gamma/2)=0$ (provided that $\lambda \in C_1 \cup C_2 \cup C_3$) and cusp points for $\cos(\gamma/2)=0$ (provided that $\lambda \in C_2$). 2.5.4. Symmetries and Maxwell strataThe phase portrait of the pendulum system (2.22) has the symmetry group $\operatorname{Sym}=\{\operatorname{Id},\varepsilon^1,\dots,\varepsilon^7\}$, described in § 2.4.3. The extension of this group to the source space of the exponential map $N=C \times \mathbb{R}_+$ was also described there. The extension of this symmetry group to the target space of the exponential map has the form
$$
\begin{equation*}
\varepsilon^i \colon g=(x,y,z) \mapsto g^i=\varepsilon^i(g)=(x^i,y^i,z^i),
\end{equation*}
\notag
$$
where
$$
\begin{equation}
\begin{aligned} \, (x^{1},y^{1},z^{1})&=(x\cosh z-y\operatorname{sinh} z, x\operatorname{sinh} z-y\cosh z,z), \\ (x^{2},y^{2},z^{2})&=(x\cosh z-y\operatorname{sinh} z, -x\operatorname{sinh} z+y\cosh z,-z), \\ (x^{3},y^{3},z^{3})&=(x,-y,-z), \\ (x^{4},y^{4},z^{4})&=(-x,y,-z), \\ (x^{5},y^{5},z^{5})&=(-x\cosh z+y\operatorname{sinh} z, x\operatorname{sinh} z-y\cosh z,-z), \\ (x^{6},y^{6},z^{6})&=(-x\cosh z+y\operatorname{sinh} z, -x\operatorname{sinh} z+y\cosh z,z), \\ (x^{7},y^{7},z^{7})&=(-x,-y,z). \end{aligned}
\end{equation}
\tag{2.23}
$$
The following analogue of Proposition 2.3 holds. Theorem 2.17. The first Maxwell time corresponding to the symmetry group $\operatorname{Sym}$ has the following expression for almost all geodesics: We also have the following result. Corollary 2.1. For each $\lambda \in C$ the first Maxwell time $t_{\rm Max}^1$ is equal to the period of oscillations of the pendulum (2.22). An analogue of Theorem 2.8 holds. 2.5.5. Estimates for the cut timeLet $p_1^1(k)\! \in\! (2K,3K)$ denote the first positive root of the equation $\operatorname{cn}p\operatorname{E}(p)- \operatorname{sn}p\operatorname{dn}p=0$. Theorem 2.18. If $\lambda \in C_1$, then $4K(k) \leqslant t_{\rm conj}^1(\lambda) \leqslant 2 p_1^1(k)$. Moreover, Theorem 2.19. If $\lambda \in C_2$, then $4k K(k) \leqslant t_{\rm conj}^1(\lambda) \leqslant 2 k p_1^1(k)$. Moreover, Theorem 2.20. If $\lambda \in C_4$, then $t_{\rm conj}^1(\lambda)=2\pi$. If $\lambda \in C_3 \cup C_5$, then $t_{\rm conj}^1(\lambda)=+\infty$. Theorem 2.21. The lower estimates for $t_{\rm conj}^1(\lambda)$ presented in Theorems 2.18 and 2.19 in the case where $\lambda \in C_1 \cup C_2$ are sharp: 2.5.6. The cut timeTheorem 2.23. (1) The function $t_{\rm cut}\colon C\to (0,+\infty]$ depends only on the energy $E$ of the pendulum (2.22). (2) The function $t_{\rm cut}$ is invariant under the vertical component of the Hamiltonian field $\vec{H}_v$ and the symmetries $\varepsilon^i \in \operatorname{Sym}$. (3) The function $t_{\rm cut}$ is continuous on $C$ and smooth on $C_1 \cup C_2$. (4) $\lim_{E \to-1}t_{\rm cut}=2 \pi$, $\lim_{E \to 1}t_{\rm cut}=+ \infty$, and $\lim_{E \to+\infty} t_{\rm cut}=0$. 2.5.7. The diffeomorphism structure of the exponential mapConsider an open dense subset of $G$ that contains no first Maxwell points: and consider its partition into connected components:
$$
\begin{equation*}
\widetilde{G}=G_1 \sqcup G_2, \quad\text{where } G_1=\{g \in G \mid z > 0\}\quad\text{and} \quad G_2=\{g \in G \mid z < 0\}.
\end{equation*}
\notag
$$
Also consider an open dense subset of the space of potentially optimal geodesics:
$$
\begin{equation*}
\widetilde N=\biggl\{(\lambda,t) \in \bigcup_{i=1}^3 N_1 \cup N_5 \Bigm| t < t_{\rm cut}(\lambda), \ \sin\frac{\gamma_{t/2}}{2} \ne 0 \biggr\}
\end{equation*}
\notag
$$
and its partition into connected components: where
$$
\begin{equation*}
D_1=\biggl\{(\lambda,t) \in \widetilde N \Bigm|\sin\frac{\gamma_{t/2}}{2} > 0 \biggr\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
D_2=\biggl\{(\lambda,t) \in \widetilde N \Bigm|\sin\frac{\gamma_{t/2}}{2} < 0\biggr\}.
\end{equation*}
\notag
$$
2.5.8. The cut locus Theorem 2.25. The cut locus $\operatorname{Cut}$ lies in the plane $\{z=0\}$. It has a partition into connected components We show the cut locus in Fig. 11. In Fig. 12 we show the cut locus and the first caustic $\operatorname{Conj}^1$. 2.5.9. SpheresThe sub-Riemannian spheres $S_R$, $R > 0$, are homeomorphic to the Euclidean 2-sphere (see the sphere $S_{\pi}$ in Fig. 13 and the sphere $S_{2 \pi}$ in Fig. 14). These spheres have singularities at the points of their intersection with the cut locus (see the intersection of $\operatorname{Cut}$ and $S_{\pi} \cap \{z < 0\}$ in Fig. 15 and the intersection of $\operatorname{Cut}$ and $S_{2\pi} \cap \{z < 0\}$ in Fig. 16). 2.5.10. The structure of optimal synthesisTheorem 2.26. (1) For each point $g_1 \in \operatorname{Cut} \setminus \operatorname{Conj}^1=\operatorname{int}_{\{z=0\}} \operatorname{Cut}$ there exist precisely two length minimizers connecting $\operatorname{Id}$ with $g_1$. For these curves $g_1$ is a cut point and a Maxwell point, but not a conjugate point. (2) For each point $g_1 \in \operatorname{Cut} \mathop{\cap} \operatorname{Conj}^1= (\partial_{\{z=0\}} \operatorname{Cut}) \setminus \{\operatorname{Id}\}$ there exists a unique length minimizer connecting $\operatorname{Id}$ with $g_1$. For it $g_1$ is a cut point and a conjugate point, but not a Maxwell point. (3) For each point $g_1 \in G \setminus(\operatorname{Cut} \cup \operatorname{Id})$ there exists a unique length minimizer connecting $\operatorname{Id}$ with $g_1$. For it $g_1$ is neither a cut point, nor a conjugate or a Maxwell point. 2.5.11. Metric linesThe metric lines through the identity element $\operatorname{Id}$ are
$$
\begin{equation*}
g(t)=\operatorname{Exp}(\lambda,t), \qquad t \in \mathbb{R}, \quad \lambda \in C_3 \cup C_5.
\end{equation*}
\notag
$$
2.5.12. Bibliographic commentsSubsection 2.5.1 is based on [159], §§ 2.5.2 and 2.5.3, on [58], §§ 2.5.4 and 2.5.5 on [59], and §§ 2.5.6–2.5.11 on [60]. 2.6. Euler’s elastic problem2.6.1. The history of the problemIn 1691 J. Bernoulli considered the problem of the shape of an elastic planar rod compressed by an external force. He deduced equations for an elastic rod which is fixed vertically to a horizontal wall and is flexed by a force applied horizontally to its upper end (a rectangular elastic):
$$
\begin{equation*}
dy=\frac{x^2\,d x}{\sqrt{1-x^4}}\,, \quad ds=\frac{d x}{\sqrt{1-x^4}}\,, \qquad x \in [0,1),
\end{equation*}
\notag
$$
where $(x,y)$ is the elastic rod and $s$ is the length parameter (the rod is deflected to a distance of 1 in the horizontal direction). Bernoulli integrated these equations by series and obtained two-sided estimates for their solution at the finite point $x=1$ [44]. In 1742 D. Bernoulli, in his letter [43] to Euler, wrote that the elastic energy of the rod is proportional to $J=\displaystyle\int\dfrac{ds}{R^2}$ , where $R$ is the curvature radius of the rod. He proposed to find the shape of elastica on the basis of the variational principle $J \to \min$. At that time Euler was writing his treatise [70] on variational calculus (which was published in 1744). He added a supplement De curvis elasticis to that book, where he applied the methods just developed to the problem of elastic rods. Euler considered a thin homogeneous elastic plate which has the form of a rectangle in its natural (stressless) state. He stated the following problem for the profile of the plate: “… among all curves of the same length which not only pass through the points A and B, but also are tangent to given straight lines at these points, that curve be determined in which the value of $\displaystyle\int_A^B\dfrac{ds}{R^2}$ is a minimum.” Euler wrote down an equation for the corresponding variational problem, which is now known as the Euler–Lagrange equation, and reduced it to the equations
$$
\begin{equation*}
dy=\frac{(\alpha+\beta x+\gamma x^2)\, dx} {\sqrt{a^4 -(\alpha+\beta x+\gamma x^2)^2}}\,,\qquad ds=\frac{a^2\,dx}{\sqrt{a^4-(\alpha+\beta x+\gamma x^2)^2}}\,,
\end{equation*}
\notag
$$
with parameters expressed in terms of the elastic characteristics and length of the rod and the load value. Using the modern language, Euler examined the qualitative behaviour of the elliptic functions parametrizing the elastic curves by means of a qualitative analysis of the equations defining these curves. Subsequently, curves describing the shape of a homogeneous planar rod were called Euler elasticae. Euler describes all types of elasticae and indicated the values of the parameters for which these types appear. He divided the elasticae into nine classes (see Figs. 19–25 below):
The first explicit parametrization of Euler’s elasticae was due to Saalschütz [122] (1880). In 1906, the future Nobel prize winner Born defended his thesis Stability of elastic curves in the plane and space [51]. He treated the problem of elasticae by methods of variational calculus and deduced from the Euler–Lagrange equations the equations
$$
\begin{equation*}
\begin{gathered} \, \dot x=\cos \theta, \quad \dot y=\sin \theta, \\ A \ddot \theta+R \sin (\theta-\gamma)=0,\quad A,R,\gamma=\operatorname{const}, \end{gathered}
\end{equation*}
\notag
$$
so that the slope angle $\theta$ of an elastica satisfies the equation of a mathematical pendulum. Then Born considered the question of the stability of elasticae with fixed endpoints and prescribed tangents at these endpoints. He proved that an arc without inflection points of an elastica is stable (on it the angle $\theta$ changes monotonically and can be taken as a parameter on the curve; Born showed that the second variation of the elastic energy functional $J=\dfrac{1}{2}\displaystyle\int \dot\theta^2\,dt$ is positive). In the general case Born presented a Jacobian vanishing at conjugate points. Since it involves quite complicated functions, Born limited himself to a numerical examination of conjugate points. He was the first to plot elasticae and verify theoretical results by experiments with elastic rods. In addition, he analyzed the stability of elasticae for various other types of boundary conditions, and obtained some results for elastic three-dimensional curves. In 1993 Euler elasticae were discovered by Jurdjevic [86] in the problem of ball rolling over a plane without twisting or slipping (see § 2.8) and by Brockett and Dai [56] in the sub-Riemannian problem on the Cartan group (see § 2.10). Euler elasticae also appear miraculously in the Martinet flat sub-Riemannian problem (see § 2.3) and in sub-Riemannian problems on $\operatorname{SE}(2)$ (see § 2.4) and the Engel group (see § 2.9). In could be interesting to understand why Euler elasticae appear in so many optimal control problems. Subsequently, Euler’s problem of elasticae was considered in [18], [111], [129]–[131], [133], [136], and [137], and our presentation in this section is based on these papers. 2.6.2. The problem statementThe mechanical setting. Assume that a uniform elastic rod on the plane $\mathbb{R}^2$ has length $l > 0$. Fix any points $a_0, a_1 \in \mathbb{R}^2$ and some unit tangent vectors $v_i \in T_{a_i} \mathbb{R}^2$, $|v_i|=1$, $i=0,1$. The problem consists in finding the profile of the rod $\gamma\colon[0,l]\to\mathbb{R}^2$, $|\dot \gamma(s)| \equiv 1$, going out of $a_0$ and coming into $a_1$ with tangent vectors $v_0$ and $v_1$, respectively:
$$
\begin{equation*}
\begin{alignedat}{2} \gamma(0)&=a_0, &\qquad \gamma(l)&=a_1, \\ \dot\gamma(0)&=v_0, &\qquad \dot\gamma(l)&=v_1, \end{alignedat}
\end{equation*}
\notag
$$
and with the minimum elastic energy where $k(s)$ is the curvature of the curve $\gamma(s)$. The optimal control problem. We introduce Cartesian coordinates $(x,y)$ in $\mathbb{R}^2$. We denote the length parameter $s$ on a curve $\gamma$ by $t$; let $t_1=l$. The curve to be determined has a parametrization $\gamma(t)=(x(t),y(t))$, $t \in [0,t_1]$, and its endpoints have coordinates $a_i=(x_i,y_i)$, $i=0,1$. Let $\theta(t)$ denote the angle between the tangent vector $\dot\gamma(t)$ and the positive direction of the $x$-axis. Finally, assume that the tangent vectors at the endpoints of $\gamma$ have coordinates $v_i=(\cos\theta_i,\sin\theta_i)$, $i=0,1$ (see Fig. 18). Then the required curve $\gamma(t)=(x(t),y(t))$ is the projection of a trajectory of the control system
$$
\begin{equation}
g =(x,y,\theta) \in M=\mathbb{R}^2_{x,y} \times S^1_{\theta},\quad u \in \mathbb{R},
\end{equation}
\tag{2.27}
$$
$$
\begin{equation}
g(0) =g_0=(x_0,y_0,\theta_0), \quad g(t_1)=g_1=(x_1,y_1,\theta_1), \quad t_1 \text{ is fixed}.
\end{equation}
\tag{2.28}
$$
For a curve $\gamma$ with natural parametrization the curvature is equal to the angular velocity: $k=\dot \theta=u$, so we obtain the quality functional A natural class of admissible controls in problem (2.24)–(2.29) is $u(\,\cdot\,) \in L^2[0, t_1]$, so an admissible trajectory satisfies $g(\,\cdot\,) \in W^{1,2}([0,t_1], M)$. In the vector notation the problems assumes the following form:
$$
\begin{equation}
\begin{gathered} \, \dot g=X_1(g)+u X_2(g),\quad g \in M=\mathbb{R}^2 \times S^1,\quad u \in \mathbb{R}, \\ \nonumber g(0)=g_0, \quad g(t_1)=g_1, \quad t_1 \text{ is fixed}, \\ \nonumber J=\frac{1}{2} \int_0^{t_1} u^2\,dt \to \min, \quad u \in L^2[0,t_1], \end{gathered}
\end{equation}
\tag{2.30}
$$
where the vector fields on the right in (2.30) are
$$
\begin{equation*}
X_1=\cos \theta\,\frac{\partial}{\partial x}+ \sin \theta\,\frac{\partial}{\partial y} \quad\text{and}\quad X_2=\frac{\partial}{\partial \theta}\,.
\end{equation*}
\notag
$$
The state space $M=\mathbb{R}^2 \times S^1$ has the natural structure of the planar motion group $G=\mathbb{R}^2 \rtimes \operatorname{SO}(2)$ (see § 2.4). Then $X_1$ and $X_2$ become left-invariant vector fields on the Lie group $G$. We presented the multiplication table for the Lie algebra $\mathfrak{g}=\mathfrak{se}(2)$ in (2.10). Thus, Euler’s elastic problem (2.24)–(2.29) is a left-invariant optimal control problem on the group $\operatorname{SE}(2)$. Hence we can assume that $g_0=\operatorname{Id}=(0,0,0)$. 2.6.3. The attainability setTheorem 2.27. The attainability set of system (2.30) from the point $\operatorname{Id}=(0,0,0)$ in time $t_1 > 0$ is Topologically, $\mathcal{A}(t_1)$ is an open solid torus (the interior of a torus) with one boundary point. Below we consider the problem of elasticae under the natural controllability condition $g_1 \in \mathcal{A}(t_1)$. 2.6.4. Existence and boundedness of optimal controlsTheorem 2.28. Let $g_1 \in \mathcal{A}(t_1)$. Then an optimal control $u \in L^2[0, t_1]$ exists; furthermore, $u \in L^{\infty}[0, t_1]$. Hence the optimal control satisfies the Pontryagin maximum principle. 2.6.5. ExtremalsAbnormal trajectories. The abnormal trajectory through the point $\operatorname{Id}$ with natural parametrization is $(x,y,\theta)\!=\!(t,0,0)$, $t \in [0,t_1]$. Its projection onto the $(x,y)$-plane is a line segment, an elastic rod in the absence of external effects. In this case the elastic energy attains its absolute minimum $J=0$, so the abnormal trajectory if optimal. This is the trajectory coming into the unique point $(t_1,0,0)$ on the boundary of the attainability set $\mathcal{A}(t_1)$. The abnormal trajectory is incidentally normal. Normal extremals. Normal extremals satisfy the Hamiltonian system
$$
\begin{equation*}
\dot \lambda=\vec{H}(\lambda),\qquad \lambda \in T^*G,
\end{equation*}
\notag
$$
where $H=h_1+h_2^2/2$ and $h_i(\lambda)=\langle \lambda,X_i\rangle$, $i=1,2,3$. In the coordinate form this system looks like
$$
\begin{equation}
\dot h_1=- h_2h_3, \quad \dot h_2=h_3, \quad \dot h_3=h_1h_2,
\end{equation}
\tag{2.31}
$$
The vertical subsystem (2.31) has a first integral, the Casimir function $F=h_1^2+h_3^2$. Consider the coordinates
$$
\begin{equation*}
c=h_2, \quad h_1=- r \cos \gamma, \quad h_2=- r \sin \gamma,
\end{equation*}
\notag
$$
in which (2.31) takes the form of a mathematical pendulum:
$$
\begin{equation}
\dot \gamma=c, \quad \dot c=-r \sin \gamma, \qquad c \in \mathbb{R}, \quad \gamma \in S^1, \quad r \equiv \operatorname{const} \geqslant 0,
\end{equation}
\tag{2.33}
$$
which is known as Kirchhoff’s kinetic analogue for elasticae. The full energy of the pendulum is Stratification of the source space of the exponential map and straightening coordinates. The exponential map for time $t_1 > 0$ in the problem of elasticae is
$$
\begin{equation*}
\operatorname{Exp}_{t_1}\colon N=\mathfrak{g}^*\to G, \qquad \lambda \mapsto \pi \circ e^{t_1 \vec{H}}(\lambda),
\end{equation*}
\notag
$$
where $\pi\colon T^*G \to G$ is the canonical projection. The source space of the exponential map $N=\mathfrak{g}^*$ is partitioned into invariant manifolds of the Hamiltonian field $\vec{H}$ by critical levels of energy $E=H$: where
$$
\begin{equation*}
\begin{gathered} \, N_1=\{\lambda \in N \mid r \ne 0, \ E \in (-r, r)\}, \\ N_2=\{\lambda \in N \mid r \ne 0, \ E \in (r,+\infty)\}, \\ N_3=\{\lambda \in N \mid r \ne 0, \ E=r, \ \gamma \ne \pi\}, \\ N_4=\{\lambda \in N \mid r \ne 0, \ E =-r \}, \\ N_5=\{\lambda \in N \mid r \ne 0, \ E =r, \ \gamma=\pi\}, \\ N_6=\{\lambda \in N \mid r=0, \ c \ne 0 \}, \end{gathered}
\end{equation*}
\notag
$$
and On $N_1$, $N_2$, and $N_3$ we introduce a coordinate system $(\varphi,k,r)$ as follows:
$$
\begin{equation*}
\begin{aligned} \, \lambda&=(\gamma,c,r) \in N_1 \quad\Longrightarrow\quad \begin{cases} \sin \dfrac{\gamma}{2}=k \operatorname{sn}(\sqrt{r}\,\varphi,k), \\ \dfrac{c}{2}=k \sqrt{r}\,\operatorname{cn}(\sqrt{r}\,\varphi,k), \\ \cos \dfrac{\gamma}{2}=\operatorname{dn}(\sqrt{r}\,\varphi,k), \end{cases} \\ &\qquad k=\sqrt{\dfrac{E+r}{2r}} \in (0,1), \quad \sqrt{r}\,\varphi\ \operatorname{mod}{4K(k)} \in [0,4K(k)]; \\ \lambda&=(\gamma,c,r) \in N_2 \quad\Longrightarrow\quad \begin{cases} \sin \dfrac{\gamma}{2}=\pm \operatorname{sn} \biggl(\dfrac{\sqrt{r}\,\varphi}{k}\,,k\biggr), \\ \dfrac{c}{2}=\pm \dfrac{\sqrt{r}}{k} \operatorname{dn}\biggl(\dfrac{\sqrt{r}\,\varphi}{k}\,,k\biggr), \\ \cos \dfrac{\gamma}{2}=\operatorname{cn} \biggl(\dfrac{\sqrt{r}\,\varphi}{k}\,,k\biggr), \end{cases} \\ &\qquad k=\sqrt{\frac{2 r}{E+r}} \in (0,1), \quad \sqrt{r}\, \varphi\ \operatorname{mod}{2K(k)k} \in [0,2K(k)k], \quad \pm=\operatorname{sign} c; \\ \lambda&=(\gamma,c,r) \in N_3 \quad\Longrightarrow\quad \begin{cases} \sin \dfrac{\gamma}{2}=\pm \tanh (\sqrt{r}\,\varphi), \\ \dfrac{c}{2}=\pm \dfrac{\sqrt{r}}{\cosh (\sqrt{r}\,\varphi)}\,, \\ \cos \dfrac{\gamma}{2}=\dfrac{1}{\cosh (\sqrt{r}\,\varphi)}\,, \end{cases} \\ &\qquad k=1, \quad \varphi \in \mathbb{R}, \quad \pm=\operatorname{sign} c. \end{aligned}
\end{equation*}
\notag
$$
A parametrization of extremals. In the domain $N_1 \cap N_2 \cup N_3$ the pendulum equation ‘straightens out’: so it has the solutions
$$
\begin{equation*}
\varphi_t=\varphi+t, \quad k,r \equiv \operatorname{const}.
\end{equation*}
\notag
$$
In the original coordinates $(\gamma,c)$ the pendulum equation (2.33) has the following solutions:
$$
\begin{equation*}
\begin{aligned} \, \lambda \in N_1 &\quad\Longrightarrow\quad \begin{cases} \sin \dfrac{\gamma_t}{2}=k_1 \operatorname{sn}(\sqrt{r}\,\varphi_t), \\ \cos \dfrac{\gamma_t}{2}=\operatorname{dn}(\sqrt{r}\,\varphi_t), \\ \dfrac{c_t}{2}=k \sqrt{r}\,\operatorname{cn}(\sqrt{r}\,\varphi_t); \end{cases} \\ \lambda \in N_2 &\quad\Longrightarrow\quad \begin{cases} \sin \dfrac{\gamma_t}{2}=\pm \operatorname{sn} \biggl(\dfrac{\sqrt{r}\,\varphi_t}{k}\biggr), \\ \cos \dfrac{\gamma_t}{2}=\operatorname{cn} \biggl(\dfrac{\sqrt{r}\,\varphi_t}{k}\biggr), \\ \dfrac{c_t}{2}=\pm \dfrac{\sqrt{r}}{k} \operatorname{dn} \biggl(\dfrac{\sqrt{r}\,\varphi_t}{k}\biggr), \end{cases} \qquad \pm=\operatorname{sign}c; \\ \lambda \in N_3 &\quad\Longrightarrow\quad \begin{cases} \sin\dfrac{\gamma_t}{2}=\pm\tanh (\sqrt{r}\,\varphi_t), \\ \cos \dfrac{\gamma_t}{2}=\dfrac{1}{\cosh (\sqrt{r}\,\varphi_t)}\,, \\ \dfrac{c_t}{2}=\pm\dfrac{\sqrt{r}}{\cosh (\sqrt{r}\,\varphi_t)}, \end{cases} \qquad \pm=\operatorname{sign}c. \end{aligned}
\end{equation*}
\notag
$$
In the degenerate cases corresponding to $\bigcup_{i=4}^7 N_i$ the pendulum equation (2.33) can be integrated by elementary functions:
$$
\begin{equation*}
\begin{alignedat}{2} \lambda \in N_4 &\quad\Longrightarrow\quad \gamma_t \equiv 0, &\qquad c_t &\equiv 0; \\ \lambda \in N_5 &\quad\Longrightarrow\quad \gamma_t \equiv \pi, &\qquad c_t &\equiv 0; \\ \lambda \in N_6 & \quad\Longrightarrow\quad \gamma_t=c t+\gamma, &\qquad c_t &\equiv c; \\ \lambda \in N_7 &\quad\Longrightarrow\quad c_t \equiv 0, &\qquad r &\equiv 0. \end{alignedat}
\end{equation*}
\notag
$$
Solutions of the horizontal subsystem (2.32) have the following parametrization. If $\lambda \in N_1$, then
$$
\begin{equation*}
\begin{aligned} \, \sin \frac{\theta_t}{2} &= k \operatorname{dn}(\sqrt{r}\,\varphi) \operatorname{sn}(\sqrt{r}\,\varphi_t)-k\operatorname{sn}(\sqrt{r}\,\varphi) \operatorname{dn}(\sqrt{r}\,\varphi_t), \\ \cos \frac{\theta_t}{2} &= \operatorname{dn}(\sqrt{r}\,\varphi) \operatorname{dn}(\sqrt{r}\,\varphi_t)+k^2\operatorname{sn}(\sqrt{r}\,\varphi) \operatorname{sn} (\sqrt{r}\,\varphi_t), \\ x_t &= \frac{2}{\sqrt r} \operatorname{dn}^2 (\sqrt r\,\varphi) \bigl(\operatorname{E}(\sqrt r\,\varphi_t)- \operatorname{E}(\sqrt r\,\varphi)\bigr) \\ &\qquad+\frac{4k^2}{\sqrt r}\operatorname{dn}(\sqrt{r}\,\varphi) \operatorname{sn}(\sqrt{r}\,\varphi) \bigl(\operatorname{cn}(\sqrt{r}\,\varphi)- \operatorname{cn} (\sqrt{r}\,\varphi_t)\bigr) \\ &\qquad+\frac{2k^2 }{\sqrt r} \operatorname{sn}^2(\sqrt{r}\,\varphi) \bigl(\sqrt r\,t+\operatorname{E}(\sqrt r\,\varphi)- \operatorname{E}(\sqrt r\,\varphi_t)\bigr)-t, \\ y_t &= \frac{2k}{\sqrt r}(2 \operatorname{dn}^2(\sqrt{r}\,\varphi)-1) \bigl(\operatorname{cn}(\sqrt{r}\,\varphi)- \operatorname{cn}(\sqrt{r}\,\varphi_t)\bigr) \\ &\qquad-\frac{2k}{\sqrt r}\operatorname{sn}(\sqrt{r}\,\varphi) \operatorname{dn}(\sqrt{r}\,\varphi) \bigl[2\bigl(\operatorname{E}(\sqrt r\,\varphi_t)- \operatorname{E}(\sqrt r\,\varphi)\bigr)-\sqrt r\,t\bigr]. \end{aligned}
\end{equation*}
\notag
$$
If $\lambda \in N_2$, then
$$
\begin{equation*}
\begin{aligned} \, \sin \frac{\theta_t}{2} &=\pm\bigl(\operatorname{cn}(\sqrt r\,\psi) \operatorname{sn}(\sqrt r\,\psi_t)-\operatorname{sn}(\sqrt r\,\psi) \operatorname{cn}(\sqrt r\,\psi_t)\bigr), \\ \cos \frac{\theta_t}{2} &=\operatorname{cn}(\sqrt r\,\psi) \operatorname{cn}(\sqrt r\,\psi_t)+\operatorname{sn}(\sqrt r\,\psi) \operatorname{sn}(\sqrt r\,\psi_t), \\ x_t &= \frac{1}{\sqrt r}\bigl(1-2 \operatorname{sn}^2(\sqrt r\,\psi)\bigr) \biggl[\frac{2}{k}\bigl(\operatorname{E}(\sqrt r\,\psi_t)- \operatorname{E}(\sqrt r\,\psi)\bigr)-\frac{2-k^2}{k^2}\sqrt r\,t\biggr] \\ &\qquad+ \frac{4}{k \sqrt r} \operatorname{cn}(\sqrt r\,\psi) \operatorname{sn}(\sqrt r\,\psi)\bigl(\operatorname{dn}(\sqrt r\,\psi)- \operatorname{dn}(\sqrt r\,\psi_t)\bigr), \\ y_t &= \pm \biggl(\frac{2}{k\sqrt r} \bigl(2\operatorname{cn}^2(\sqrt r\,\psi)-1\bigr) \bigl(\operatorname{dn}(\sqrt r\,\psi)-\operatorname{dn}(\sqrt r\,\psi_t)\bigr) \\ &\qquad- \frac{2}{\sqrt r} \operatorname{sn}(\sqrt r\,\psi) \operatorname{cn}(\sqrt r\,\psi)\biggl[\frac{2}{k} \bigl(\operatorname{E}(\sqrt r\,\psi_t)-\operatorname{E}(\sqrt r\,\psi)\bigr)- \frac{2-k^2}{k^2}\sqrt r\,t\biggr]\biggr), \end{aligned}
\end{equation*}
\notag
$$
where $\pm=\operatorname{sign} c$ and $\psi_t=\varphi_t/k=(\varphi+t)/k$. If $\lambda \in N_3$, then
$$
\begin{equation*}
\begin{gathered} \, \begin{aligned} \, \sin \frac{\theta_t}{2} &= \pm \biggl(\frac{\tanh (\sqrt r\,\varphi_t)}{\cosh (\sqrt r\,\varphi)}- \frac{\tanh (\sqrt r\,\varphi)} {\cosh (\sqrt r\,\varphi_t)}\biggr), \\ \cos \frac{\theta_t}{2} &= \frac{1}{\cosh (\sqrt r\,\varphi) \cosh (\sqrt r\,\varphi_t)}+\tanh (\sqrt r\,\varphi) \tanh (\sqrt r\,\varphi_t), \end{aligned} \\ \begin{aligned} \, x_t &=(1-2\tanh^2(\sqrt r\,\varphi)) t +\frac{4\tanh (\sqrt r\,\varphi)} {\sqrt r\,\cosh (\sqrt r\,\varphi)} \biggl(\frac{1}{\cosh (\sqrt r\,\varphi)} -\frac{1}{\cosh (\sqrt r\,\varphi_t)}\biggr), \\ y_t &=\pm\biggl[\frac{2}{\sqrt r} \biggl(\frac{2}{\cosh^2(\sqrt r\,\varphi)}-1\biggr) \biggl(\frac{1}{\cosh (\sqrt r\,\varphi)} -\frac{1}{\cosh (\sqrt r\,\varphi_t)}\biggr)\biggr. -2\,\frac{\tanh (\sqrt r\,\varphi)} {\cosh (\sqrt r\,\varphi)}\,t \biggr], \end{aligned} \end{gathered}
\end{equation*}
\notag
$$
where $\pm=\operatorname{sign} c$. If $\lambda \in N_4 \cup N_5 \cup N_7$, then If $\lambda \in N_6$, then
$$
\begin{equation*}
\theta_t=ct, \quad x_t=\frac{\sin (ct)}{c}\,, \quad y_t=\frac{1-\cos (ct)}{c}\,.
\end{equation*}
\notag
$$
Euler elasticae. The projections of extremal trajectories onto the $(x,y)$-plane are Euler elasticae. Their equations are
$$
\begin{equation}
\begin{gathered} \, \dot x=\cos \theta, \quad \dot y=\sin \theta, \nonumber \\ \ddot \theta=-r \sin(\theta-\gamma), \qquad r,\gamma \equiv \operatorname{const}. \end{gathered}
\end{equation}
\tag{2.34}
$$
Depending on the energy of the pendulum $E=\dot\theta^2/2-r \cos(\theta-\gamma) \in [-r,+\infty)$ and the value of the Casimir function $r \geqslant 0$, elasticae have qualitatively different types, which were discovered by Euler. If the energy $E$ takes the minimum value $-r < 0$, so that $\lambda \in N_4$, then the elastica $(x_t,y_t)$ is a straight line. The corresponding motion of the pendulum (2.34) (Kirchhoff’s kinetic analogue) is staying in stable equilibrium. If $E \in (-r,r)$, $r > 0$, so that $\lambda \in N_1$, then the pendulum (2.34) oscillates between the extremal values of the angle and the angular velocity $\dot\theta$ changes sign. The corresponding elasticae have inflection points for $\dot \theta=0$ and vertices for $|\dot\theta|=\max$, because $\dot \theta$ is the curvature of the elastic. Such elasticae are said to be inflectional (see Figs. 19–23). Different cases in these figures depend on the value of the modulus of elliptic functions $k=\sqrt{E+r}/(2r) \in (0,1)$:
$$
\begin{equation*}
\begin{aligned} \, k \in \biggl(0,\frac{1}{\sqrt 2}\biggr) &\quad\Longrightarrow\quad \text{Fig. 19}, \\ k=\frac{1}{\sqrt 2} &\quad\Longrightarrow\quad \text{Fig. 20}, \\ k \in \biggl(\frac{1}{\sqrt 2}\,,k_0\biggr) &\quad\Longrightarrow\quad \text{Fig. 21}, \\ k=k_0 &\quad\Longrightarrow\quad \text{Fig. 22}, \\ k \in \biggl(k_0,1\biggr) &\quad\Longrightarrow\quad \text{Fig. 23}. \end{aligned}
\end{equation*}
\notag
$$
The value $k=1/\sqrt 2$ corresponds to the rectangular elastica, considered by J. Bernoulli (see § 2.6.1): see Fig. 20. The value $k \approx 0.909$ corresponds to a periodic elastica of the form of a figure-of-eight: see Fig. 22. As Euler noted, for $k \to 0$ inflectional elasticae are similar to sinusoids, which corresponds to a harmonic oscillator $\ddot \theta=- r (\theta-\gamma)$ as Kirchhoff’s kinetic analogue; see Fig. 19. If $E=r > 0$ and $\theta-\gamma \ne \pi$, so that $\lambda \in N_3$, then the pendulum (2.34) tends to an unstable equilibrium ($\theta-\gamma=\pi$, $\dot \theta=0$) along a separatrix of a saddle, and the corresponding elastica (an ‘Euler soliton’) makes one loop: see Fig. 24. If $E=r > 0$ and $\theta-\gamma=\pi$, that is, $\lambda \in N_5$, then the pendulum (2.34) is in an unstable equilibrium ($\theta-\gamma=\pi$, $\dot \theta=0$) and the elastica is a straight line. If $E > r > 0$, so that $\lambda \in N_2$, then Kirchhoff’s kinetic analogue is the pendulum (2.34) rotating anticlockwise (for $\dot \theta > 0$) or clockwise (for $\dot \theta < 0$). The corresponding elasticae have a non-zero curvature $\dot \theta$ and no inflection points; they are said to be non-inflectional: see Fig. 25. If $r=0$ and $\dot \theta \ne 0$, so that $\lambda \in N_6$, then the pendulum (2.34) rotates uniformly in zero gravity. The corresponding elastica is a circle. Finally, if $r=0$ and $\dot \theta=0$, so that $\lambda \in N_7$, then the pendulum (2.34) rests in zero gravity (so that the equilibrium is unstable) and the elastica is a straight line. In the pictures of elasticae in Figs. 19–25 the ratio $x/y$ is not always correctly reproduced for reasons of space. Periodic motions of the pendulum (2.33) (or (2.34)) have the period
$$
\begin{equation*}
T=\begin{cases} 4\,\dfrac{K(k)}{\sqrt r}\,, & \lambda \in N_1, \\ 2\,\dfrac{k K(k)}{\sqrt r}\,, & \lambda \in N_2, \\ \dfrac{2\pi}{|c|}\,, & \lambda \in N_6. \end{cases}
\end{equation*}
\notag
$$
2.6.6. Symmetries and Maxwell strataThe phase portrait of the pendulum (2.33) is preserved by the symmetry group $\operatorname{Sym}$, which is generated by the reflection $\varepsilon^1$ in the $\gamma$-axis, the reflection $\varepsilon^2$ in the $c$-axis, and the reflection $\varepsilon^3$ in the origin $(\gamma,c)=(0,0)$:
$$
\begin{equation*}
\operatorname{Sym}= \{\operatorname{Id},\varepsilon^1,\varepsilon^2,\varepsilon^3\} \simeq \mathbb{Z}_2 \times \mathbb{Z}_2.
\end{equation*}
\notag
$$
These symmetries extend in a natural way to the source space $N=\mathfrak{g}^*$ and the target space $G$ of the exponential map $\operatorname{Exp}_t$. If $\nu=(\gamma,c,r) \in N$, then where
$$
\begin{equation*}
(\gamma^1,c^1)=(\gamma_t,-c_t), \quad (\gamma^2,c^2)=(-\gamma_t, c_t), \quad (\gamma^3,c^3)=(-\gamma,-c).
\end{equation*}
\notag
$$
If $g=(x,y,\theta) \in G$, then where
$$
\begin{equation*}
\begin{aligned} \, (x^1, y^1, \theta^1)&= (x \cos \theta+y \sin \theta,-x \sin \theta+y \cos \theta,-\theta), \\ (x^2, y^2, \theta^2)&= (x \cos \theta+y \sin \theta, x \sin \theta-y \cos \theta, \theta), \\ (x^3, y^3, \theta^3)&=(x,-y,-\theta). \end{aligned}
\end{equation*}
\notag
$$
Proposition 2.4. The group $\operatorname{Sym}= \{\operatorname{Id},\varepsilon^1,\varepsilon^2,\varepsilon^3\}$ consists of the symmetries of the exponential map. Theorem 2.29. For almost all extremal trajectories $g_t=\operatorname{Exp}_t(\lambda)$, $\lambda \in N$, the first Maxwell time corresponding to the symmetry group $\operatorname{Sym}$ has the following expression: An observation similar to Remark 2.1 and a result on invariant properties of the function $t_{\rm Max}^1\colon N\to(0,+\infty]$ which is similar to Theorem 2.8 are true. 2.6.7. Estimates for the conjugate timeFor Euler elasticae, a very important question from the point of view of applications is their local optimality since it means the stability of the elastica under small perturbations of the profile such that the endpoints and tangents at these endpoints are fixed. From the theoretic point of view answering this question is a step towards the investigation of the global optimality of elasticae. Theorem 2.30. Let $\lambda=(k,\varphi,r) \in N_1$. Then the first conjugate time $t_{\rm conj}^1(\lambda)$ on the trajectory $\operatorname{Exp}_t(\lambda)$ belongs to the closed interval with endpoints $4 K(k)/\sqrt r$ and $2 p_1(k)/\sqrt r$, namely, Corollary 2.2. Let $\lambda=(k,\varphi,r) \in N_1$. Then Corollary 2.3. Let $\lambda=(k,\varphi,r) \in N_1$ and $t_1 > 0$, and let be an arc of the corresponding elastica. (1) If $\Gamma$ contains no inflection points, then it is locally optimal. (2) If $k \in (0,k_0]$ and $\Gamma$ contains precisely one inflection point, then it is locally optimal. (3) If $\Gamma$ contains at least three interior inflection points, then it is not locally optimal. Consider arcs of inflectional elasticae (2.35) with midpoint at a vertex, that is, assume that the local extremum of the curvature of the elastica is attained at $(x_{t_1/2},y_{t_1/2})$. For examples of such arcs, see Fig. 26. Set $t_1^1=(2/\sqrt r\,)p_1(k)$, where the function $p_1(k)$ was defined in Theorem 2.29. Now we consider arcs of inflectional elasticae (2.35) centred at an inflection point, that us, we assume that the elastica has zero curvature at $(x_{t_1/2},y_{t_1/2})$. For examples of such arcs, see Fig. 27. Theorem 2.33. Let $\lambda \in N_2 \cup N_3 \cup N_6$. Then the extremal trajectory $g(t)=\operatorname{Exp}_t(\lambda)$ contains no conjugate points for $t > 0$. Thus, if an arc of an elastica contains no inflection points, then it is stable; if it contains at least three interior inflection points, then it is unstable. If there are one or two inflection points, then the elastica can be stable or unstable alike. 2.6.8. The diffeomorphism structure of the exponential mapLet $t_1=1$, $\operatorname{Exp}=\operatorname{Exp}_1$, and
$$
\begin{equation*}
\mathcal{A}=\mathcal{A}_1=\{(x,y,\theta) \in G \mid x^2+y^2 < 1 \text{ or } (x,y,\theta)=(1,0,0)\}.
\end{equation*}
\notag
$$
The case of general $t_1 > 0$ reduces to $t_1=0$ by homotheties of the $(x,y)$-plane:
$$
\begin{equation*}
(x,y,\theta,t,u,t_1,J) \mapsto (\tilde x,\tilde y,\tilde\theta,\tilde t, \tilde u,\tilde t_1,\tilde J)=(e^sx,e^sy,\theta,e^st,e^{-s}u,e^st_1,e^{-s}J).
\end{equation*}
\notag
$$
Consider a subset of $\mathcal{A}$ containing no fixed points of the reflections $\varepsilon^1$ and $\varepsilon^2$:
$$
\begin{equation*}
\begin{gathered} \, \widetilde{G}=\{g \in \mathcal{A} \mid \varepsilon^i (g) \ne g, \ i=1, 2\}= \biggl\{g \in \mathcal{A} \Bigm| \sin\biggl(\frac{\theta}{2}\biggr)P(g) \ne 0 \biggr\}, \\ P(g)=x \sin \frac{\theta}{2}-y \cos \frac{\theta}{2}\,, \end{gathered}
\end{equation*}
\notag
$$
and consider its partition into connected components where
$$
\begin{equation*}
G_{\pm}=\{ g \in G \mid \theta \in (0, 2\pi), \ x^2+y^2 < 1, \ \operatorname{sign} P(g)=\pm 1 \}.
\end{equation*}
\notag
$$
Also consider the open dense subset of the space of potentially optimal extremal trajectories
$$
\begin{equation*}
\widetilde{N}=\biggl\{\lambda \in \bigcup_{i=1}^3 N_i \Bigm| t_1 < t_{\rm Max}^1(\lambda), \ c_{t_1/2} \sin\gamma_{t_1/2} \ne 0\biggr\},
\end{equation*}
\notag
$$
and its connected components where
$$
\begin{equation*}
\begin{gathered} \, D_1=\biggl\{\lambda \in \bigcup_{i=1}^3 N_i \Bigm| t_1 < t_{\rm Max}^1(\lambda), \ c_{t_1/2} > 0, \ \sin \gamma_{t_1/2} > 0\biggr\}, \\ D_2=\biggl\{\lambda \in \bigcup_{i=1}^3 N_i \Bigm| t_1 < t_{\rm Max}^1(\lambda), \ c_{t_1/2} < 0, \ \sin \gamma_{t_1/2} > 0\biggr\}, \\ D_3=\biggl\{\lambda \in \bigcup_{i=1}^3 N_i \Bigm| t_1 < t_{\rm Max}^1(\lambda), \ c_{t_1/2} < 0, \ \sin \gamma_{t_1/2} < 0\biggr\}, \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
D_4=\biggl\{\lambda \in \bigcup_{i=1}^3 N_i \Bigm| t_1 < t_{\rm Max}^1(\lambda), \ c_{t_1/2} > 0, \ \sin \gamma_{t_1/2} < 0\biggr\}.
\end{equation*}
\notag
$$
Corollary 2.4. The map $\operatorname{Exp}\colon\widetilde{N}\to \widetilde{G}$ is a two-sheeted cover. 2.6.9. Optimal elasticae for various boundary conditionsGeneric boundary conditions. If $g_1 \in G_+$, then there exists a unique pair $(\lambda_1,\lambda_3) \in D_1 \times D_3$ such that $\operatorname{Exp}(\lambda_1)=\operatorname{Exp}(\lambda_3)=g_1$. The optimal trajectory is either $q^1(t)=\operatorname{Exp}_t(\lambda_1)$ or $q^3(t)=\operatorname{Exp}_t(\lambda_3)$, $t \in [0,1]$. As the optimal trajectory we must take the one at which the quality functional $J[q^i(\,\cdot\,)]=\dfrac{1}{2}\displaystyle\int_0^1 (c_t^i)^2\,dt$ takes the smaller value. If $J[q^1(\,\cdot\,)]=J[q^3(\,\cdot\,)]$, then both trajectories are optimal (we show this case in Fig. 28). If $g_1 \in G_-$, then we select optimal elasticae in a similar way among the ones corresponding to the covectors $\lambda_2 \in D_2$ and $\lambda_4 \in D_4$ for which $\operatorname{Exp}(\lambda_2)=\operatorname{Exp}(\lambda_4)= g_1$. The case $(x_1,y_1,\theta_1)=(1,0,0)$. The optimal elastica is the line segment $(x,y)=(t,0)$, $t \in [0,1]$. The case $x_1>0$, $y_1=0$, $\theta_1=\pi$. In this case $g_1 \in G_+$ and the equation $\operatorname{Exp}(\lambda)=g_1$, $\lambda \in \widetilde{G}$, has too roots, $\lambda_1 \in D_1$ and $\lambda_3 \in D_3$. The trajectories $q^1(t)=\operatorname{Exp}_t(\lambda_1)$ and $q^3(t)=\operatorname{Exp}_t(\lambda_3)$ correspond to the same value of $J$, so both of them are optimal. The corresponding optimal inflection elasticae are mirror symmetric with respect to the $x$-axis (see Fig. 29). The case $x_1<0$, $y_1=0$, $\theta_1=\pi$. This is similar to the previous case (see Fig. 30). The case $x_1=0$, $y_1=0$, $\theta_1=\pi$. The only optimal elastica, a ‘water drop’, is defined by the parameters $\lambda=(\varphi,k,r) \in N_1$, where $\varphi=\dfrac{\tau}{2p}-\dfrac{1}{2}$ , $r=4 p^2$, $\operatorname{sn}\tau=0$, $1-2k^2\operatorname{sn}^2 p=0$, and $2\operatorname{E}(p)-p=0$ (see Fig. 31). The case $x_1=0$, $y_1=0$, $\theta_1=0$. There exist two optimal elasticae, which are circles mirror symmetric with respect to the $x$-axis. The case $x_1>0$, $y_1=0$, $\theta_1=0 $. There are two or four optimal elasticae: there exists $x_* \in (0.4,0.5)$ such that
The case $x_1<0$, $y_1=0$, $\theta_1=0$. There are two optimal non-inflectional elasticae (see Fig. 33). 2.6.10. Bibliographic commentsThe first classical study of elasticae was presented by Euler in [70]. Subsection 2.6.1 concerning the history of the problem of elasticae is based on the classical sources [105], [153], and [154]. A remarkable presentation of this history was given in [99]. Note that the problem of elasticae had long been of purely theoretical interest as an example of the use of the theory of elliptic functions (see, for instance, [78] and [105]). In connection with widespread introduction of steel in project design and construction of flexible thin-walled structures, which encouraged the development of the stability theory of deformed systems, the solution of the problem of elasticae became a question of practical importance. In particular, the following questions arose, which are important for applications to engineering: what is the behaviour of a compressed rack under loads higher than the Euler critical value, what is the shape of the rack in this case, is it unique and stable? A number of papers (see [49], [69], [75], [96], [119], [148], [149], [161]) were concerned with these questions, considering various conditions for supporting and loading inextensible elastic rods. During the last decades the interest to elasticae has grown because of applications of the theory of flexible rods to the analysis of micro- and nanostructures in biology and nanotechnology [76], [84], [115], [151]. It has been confirmed that there exists a variety of equilibrium shapes for a fixed load. Subsections 2.6.2–2.6.6 are based on [130], § 2.6.7 is based on [131] and [133], and §§ 2.6.8 and 2.6.9 on [137] and [136]. Problems of elasticae were also considered in [2], [9], [12], [79], [87]–[89], [106], [107], [117], and [142]. 2.7. A left-invariant sub-Riemannian problem of general form on the group $\operatorname{SO}(3)$2.7.1. The problem statementIt follows from the classification of contact left- invariant sub-Riemannian structures on three-dimensional Lie groups [1] that, given an arbitrary structure of this type on the group $G=\operatorname{SO}(3)$, we can find an orthonormal frame $(X_1,X_2)$ with multiplication table
$$
\begin{equation}
[X_2,X_1]=X_3, \quad [X_1,X_3]=(\kappa+\chi)X_2, \quad [X_2,X_3]=(\chi-\kappa)X_1,
\end{equation}
\tag{2.36}
$$
where $\kappa$ and $\chi$, $\kappa \geqslant \chi \geqslant 0$, are differential invariants of the sub-Riemannian structure. A uniform extension of $X_1$ and $X_2$ changes the distance function and both invariants $\kappa$ and $\chi$ in the same proportion. In [1] the normalization $\kappa^2+\chi^2=1$ was used. In this section it is more convenient to assume that $\kappa+\chi=1$ and use the invariant $a=\sqrt{2\chi} \in [0,1)$. The case $a=0$ corresponds to the axially symmetric sub-Riemannian structure considered in [55]. The following vector fields satisfy the multiplication table (2.36):
$$
\begin{equation*}
X_1(g)=L_{g*} A_2, \quad X_2(g)=\sqrt{1-a^2}\,L_{g*} A_1, \quad X_3(g)=\sqrt{1-a^2}\,L_{g*} A_3,
\end{equation*}
\notag
$$
where the basis $A_1$, $A_2$, $A_3$ of the Lie algebra $\mathfrak{g}=\mathfrak{so}(3)$ has the form
$$
\begin{equation}
A_1=\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix},\quad A_2=\begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix},\quad A_3=\begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}.
\end{equation}
\tag{2.37}
$$
2.7.2. A parametrization of geodesicsAbnormal extremal trajectories are constant in time. To parametrize normal geodesics consider the Hamiltonians
$$
\begin{equation*}
h_i(\lambda)=\langle \lambda,X_i(g)\rangle,\quad i=1,2,3,\qquad H=\frac{1}{2}(h_1^2+h_2^2).
\end{equation*}
\notag
$$
Extremals with normal parametrization are themselves parametrized by points in the cylinder $C=\mathfrak{g}^* \cap \{H=1/2\}$. We introduce the coordinates $(\psi,c)$ on this cylinder: The normal Hamiltonian system of the Pontryagin maximum principle has the form
$$
\begin{equation}
\dot h_1=h_2 h_3, \quad \dot h_2=- h_1 h_3, \quad \dot h_3=a^2 h_1 h_2,
\end{equation}
\tag{2.38}
$$
The vertical subsystem (2.38) defines the pendulum equation
$$
\begin{equation}
\dot \psi=c, \quad \dot c=-\frac{a^2}{2} \sin(2\psi)
\end{equation}
\tag{2.40}
$$
on $C$. The cylinder is stratified by invariant sets of the system (2.40), which are defined in terms of the full energy $E=2c^2-a^2\cos(2\psi)$ of the pendulum:
$$
\begin{equation*}
\begin{gathered} \, C_1=\{\lambda\in C\mid E\in(-a^2,a^2)\}\quad (\text{the region inside the separatrices}), \\ C_2=\{\lambda\in C\mid E\in(a^2, +\infty)\}\quad (\text{the region outside the separatrices}), \\ C_3=\{\lambda\in C\mid E=a^2, c \ne 0\}\quad (\text{the separatrices}), \\ C_4=\{\lambda\in C\mid E=-a^2\}\quad (\text{the stable equilibrium}), \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
C_5=\{\lambda\in C\mid E=a^2, c=0\}\quad (\text{the unstable equilibrium}).
\end{equation*}
\notag
$$
On $C_1$, $C_2$, and $C_3$ we introduce coordinates $(\theta,k)$ which straighten the equation (2.40). In the domain $C_1$ we have
$$
\begin{equation*}
\begin{gathered} \, \sin\psi=s_1k\operatorname{sn}(a\theta,k), \qquad \cos\psi=s_1\operatorname{dn}(a\theta,k), \qquad c=ak\operatorname{cn}(a\theta,k), \\ s_1=\operatorname{sign}\cos\psi,\qquad k=\sqrt{\frac{E+a^2}{2a^2}}\in(0,1),\qquad \theta\in\biggl[0,\frac{4K(k)}{a}\biggr]. \end{gathered}
\end{equation*}
\notag
$$
In the domain $C_2$,
$$
\begin{equation*}
\begin{gathered} \, \sin\psi=s_2\operatorname{sn}\biggl(\frac{a\theta}{k}\,,k\biggr),\qquad \cos\psi=\operatorname{cn}\biggl(\frac{a\theta}{k}\,,k\biggr), \qquad c=\frac{s_2a}{k}\operatorname{dn}\biggl(\frac{a\theta}{k}\,,k\biggr), \\ s_2=\operatorname{sign}c,\qquad k=\sqrt{\frac{2a^2}{E+a^2}}\in (0,1),\qquad \theta \in\biggl[0,\frac{4kK(k)}{a}\biggr]. \end{gathered}
\end{equation*}
\notag
$$
On the set $C_3$,
$$
\begin{equation*}
\sin\psi=s_1s_2\tanh(a\theta),\ \ \cos\psi=\frac{s_1}{\cosh(a\theta)}\,,\ \ c=\frac{s_2a}{\cosh(a\theta)}\,, \ \ \theta \in(-\infty,+\infty), \ \ k=1.
\end{equation*}
\notag
$$
Then for $(\psi_0,c_0) \in C_1 \cup C_2 \cup C_3$ the pendulum equation has the solution $\theta(t)=t+\theta_0$, $k \equiv \operatorname{const}$. For $(\psi_0,c_0) \in C_4$ we have $\psi \equiv \pi n$, $n \in \mathbb{Z}$, $c=0$, while for $(\psi_0,c_0) \in C_5$ we have $\psi \equiv-\pi/2+\pi n$, $n \in \mathbb{Z}$, $c=0$. To parametrize the solutions of the horizontal subsystem (2.39) we represent them in terms of the Euler angles:
$$
\begin{equation*}
g_t=\exp(-\varphi_1(0) A_3) \exp(-\varphi_2(0) A_1) \exp(\varphi_3(t) A_3) \exp(\varphi_2(t) A_1) \exp(\varphi_1(t) A_3).
\end{equation*}
\notag
$$
Then
$$
\begin{equation}
\cos \varphi_2 =\frac{c}{\sqrt M}\,, \qquad \sin \varphi_2 =\sqrt{\frac{M-c^2}{M}}\,,
\end{equation}
\tag{2.41}
$$
$$
\begin{equation}
\cos \varphi_1 =\frac{h_1\sqrt{1-a^2}}{\sqrt{M-c^2}}\,, \qquad \sin \varphi_1 =\sqrt{\frac{h_2}{M-c}}\,,
\end{equation}
\tag{2.42}
$$
where $M=h_2^2+(1-a^2) h_1^2+c^2$ is a first integral of the subsystem (2.38). The angle $\varphi_3$ satisfies the equation
$$
\begin{equation}
\dot\varphi_3=\frac{\sqrt{M (1-a^2)}}{M-c^2}= \frac{\sqrt{M(1-a^2)}}{1-a^2 h_1^2};
\end{equation}
\tag{2.43}
$$
it is a monotone function of time because
$$
\begin{equation*}
0 < \sqrt{M(1-a^2)} \leqslant \dot \varphi_3 \leqslant \sqrt{\frac{M}{1-a^2}}\,.
\end{equation*}
\notag
$$
Solutions of this equation have the following form:
2.7.3. Periodic geodesicsProposition 2.5. For each $a \in (0,1)$ there exist infinitely many geodesics in the corresponding sub-Riemannian problem on $\operatorname{SO}(3)$. For $\lambda \in C_1$ (or $\lambda \in C_2$) the relevant geodesic can only have period $T=4K(k)/a$ ($T=4k K(k)/a$, respectively); such trajectories exist if and only if $\varphi_3(m T)=2 \pi n$ for some $n,m \in \mathbb{N}$. This equality holds along a geodesic if and only if in the case of $C_1$ or in the case of $C_2$. Different irreducible fractions $n/m \in \mathbb{Q}_+$ correspond to different periodic geodesics.Proposition 2.6. Every periodic geodesic for $\lambda \in C_1$ (or $\lambda \in C_2$) is uniquely defined by an irreducible fraction $n/m \in \mathbb{Q}_+$ satisfying (2.44) (respectively, (2.45)). For $\lambda \in C_3$ geodesics are aperiodic. For $\lambda \in C_4 \cup C_5$ geodesics are periodic. Since $\pi_1(\operatorname{SO}(3))=\mathbb{Z}_2$, there exist only two homotopy classes of closed paths on $\operatorname{SO}(3)$. The following result shows which periodic geodesics are contractible (null homotopic). Proposition 2.7. Consider a periodic geodesic $g_t \in \operatorname{SO}(3)$ which is the projection of an extremal $\lambda_t \in C_1$ (or $\lambda_t \in C_2$) and is defined by an irreducible fraction $n/m \in \mathbb{Q}_+$ satisfying (2.44) (respectively, (2.45)). Then $g_t$ is reducible if and only if $n$ is even. All geodesics corresponding to $\lambda \in C_4 \cup C_5$ are non-contractible. 2.7.4. Conditions for optimalityConsider the unit 3-sphere in the algebra of quaternions
$$
\begin{equation*}
S^3=\{q=q^0+i q^1+j q^2+k q^3 \in \mathbb{H} \mid (q^0)^2+(q^1)^2+(q^2)^2+(q^3)^2=1\}.
\end{equation*}
\notag
$$
It is simply connected and covers the group $\operatorname{SO}(3)$ with multiplicity two. The geodesic $g_t \in \operatorname{SO}(3)$ lifts to a curve $q_t \in S^3$, $q_0=1$, of the form
$$
\begin{equation*}
q_t=\exp\biggl(-\frac{\varphi_1(0)}{2}k\biggr) \exp\biggl(-\frac{\varphi_2(0)}{2}i\biggr) \exp\biggl(\frac{\varphi_3(t)}{2}k\biggr) \exp\biggl(\frac{\varphi_2(t)}{2}i\biggr) \exp\biggl(\frac{\varphi_1(t)}{2}k\biggr),
\end{equation*}
\notag
$$
where the Euler angles $\varphi_i(t)$ coincide with the analogous angles in (2.41)–(2.43). Theorem 2.35. Let $g_t \in \operatorname{SO}(3)$, $t \in [0,t_1]$, be a geodesic, and let $q_t \in S^3$, $q_0=1$, be its lift to $S^3$. Let $\theta_t$ be the corresponding straightened coordinate variable of the pendulum system (2.40), and let $\tau=a(\theta_0+t/2)$. Then $g_t$ is not optimal, provided that one of the following conditions holds for some $t \in (0,t_1)$: (1) $q_t^0=0$; (2) $q_t^1=0$ and $\operatorname{sn} \tau \ne 0$ for $\lambda_0 \in C_1 \cup C_2$, or $\tau \ne 0$ for $\lambda_0 \in C_3$; (3) $q_t^2=0$ and $\operatorname{cn} \tau \ne 0$ for $\lambda_0 \in C_1$; (4) $q_t^3=0$ and $\operatorname{cn} \tau \ne 0$ for $\lambda_0 \in C_2$. 2.7.5. Bibliographic commentsThe results presented in this section were obtained in [46]. 2.8. The problem of a ball rolling over a plane without twisting or slipping2.8.1. The history of the problemIn 1983 Hammersley [83] considered the following Oxford ball problem. A unit ball lies on an infinite horizontal plane. The state of the ball is determined by its orientation in space and its position on the plane. We must take the ball from the given initial state to a prescribed terminal state by a sequence of rollings. Each rolling proceeds along a straight line on the plane: we select the distance and direction of each rolling, but there must be no twisting or slipping, that is, the axis of rotation must remain horizontal and the velocity of the ball at the point of contact with the plane must be equal to zero. What minimum number of rollings $N$ do we require to attain any terminal state? With the help of quaternions Hammersley showed that $N \in \{3,4\}$. Next he stated two continuous versions of the ball problem:
In the final section of [83], entitled “Variants for the twenty-first century” Hammersley stated a number of variants and generalizations of these problems, which are still open. In 1986, Arthgurs and Walsh [29] examined problem (a). Using quaternions and the Pontryagin maximum principle they showed that the point of contact between the ball and the $(x,y)$-plane satisfies the equations
$$
\begin{equation*}
\begin{gathered} \, \dot x=\sin \psi, \quad \dot y=-\cos \psi, \\ \ddot \psi=\lambda \cos(\psi+\varepsilon), \qquad \lambda,\varepsilon \equiv \operatorname{const}. \end{gathered}
\end{equation*}
\notag
$$
They observed that these equations are integrable by elliptic integrals of the first and third kind, and left the problem of optimal control to numerical investigation. Independently of these papers, in 1993 Brockett and Dai [56] stated a ‘plate-ball problem’. They considered a ball which rolls without twisting or slipping between two horizontal plates which are placed at a distance equal to the diameter of the ball between them. They wrote out a control system of the form (2.46)–(2.50) (see below) for the ball and showed that a nilpotent approximation to this system is equivalent to the control system (2.117) on the Cartan group (see § 2.10). In the same year of 1993 Jurdjevic [86] considered thoroughly the problem of optimal rolling of a ball over a plane without twisting or slipping, where he relied on the statement due to Brockett and Dai [56] and was independent of [29] and [83]. Jurdjevic treated this problem as a left-invariant control problem on the Lie group $G=\mathbb{R}^2\times\operatorname{SO}(3)$:
$$
\begin{equation}
\dot R=R\begin{pmatrix} 0 & 0 &-u_1\\ 0 & 0 & -u_2\\ u_1 & u_2 & 0 \end{pmatrix},
\end{equation}
\tag{2.47}
$$
$$
\begin{equation}
g=(x,y,R) \in G, \quad u=(u_1,u_2) \in \mathbb{R}^2,
\end{equation}
\tag{2.48}
$$
$$
\begin{equation}
g(0)=\operatorname{Id}=(0,0,E_{11}+E_{22}+E_{33}), \qquad g(t_1)=g_1 ,
\end{equation}
\tag{2.49}
$$
$$
\begin{equation}
J=\frac{1}{2} \int_0^{t_1}(u_1^2+u_2^2)\,dt \to \min.
\end{equation}
\tag{2.50}
$$
Next he used the Pontryagin maximum principle in the invariant form for Lie groups (see [6] and [88]) and obtained the following results. The optimal abnormal controls are constant in time, they result in rolling along a straight line; these controls are non-strictly abnormal. Normal extremals are trajectories of the Hamiltonian system with Hamiltonian where the Hamiltonians $h_1$ and $h_2$ correspond to the vector fields $\partial/\partial x$ and $\partial/\partial y$, while $H_1$, $H_2$, and $H_3$ correspond to the left-invariant fields on $\operatorname{SO}(3)$ with generators
$$
\begin{equation*}
A_1=E_{32}-E_{23}, \qquad A_2=E_{13}-E_{31}, \quad\text{and}\quad A_3=E_{21}-E_{12},
\end{equation*}
\notag
$$
defining rotations of three-dimensional space. The vertical subsystem of this Hamiltonian system is as follows:
$$
\begin{equation*}
\begin{aligned} \, \dot h_1&=\dot h_2=0, \\ \dot H_1&=(h_1-H_2) H_3, \\ \dot H_2&=(h_2+H_1)H_3, \\ \dot H_3&=-h_1 H_1-h_2 H_2. \end{aligned}
\end{equation*}
\notag
$$
This subsystem has the integrals $h_1$, $h_2$, $H$, and $M=H_1^2+H_2^2+H_3^2$, so it is integrable. In addition, it is reduced to a pendulum equation. To integrate the equations for the orientation $R(t) \in\operatorname{SO}(3)$ of the ball, the Euler angles $\varphi_1$, $\varphi_2$, and $\varphi_3$ are introduced; differential equations for these angles are obtained, qualitatively investigated, and integrated in part. It is shown that the trajectory of the point of contact $(x(t),y(t))$ between the ball and the plane is an Euler elastica (see § 2.6). A connection between the type of the intersection between the cylinder $\{H=\operatorname{const}\}$ and the sphere $\{M=\operatorname{const}\}$, the type of the elastica, and the qualitative behaviour of the Euler angles $\varphi_1$, $\varphi_2$, and $\varphi_3$ is discovered. The rest of this section is based on [114] and [132]. 2.8.2. The statement of the problemA mechanical setting. Consider the mechanical system of two horizontal planes and a sphere tangent to these planes. The lower plane is fixed, and the sphere rolls over it without twisting or slipping, driven by the horizontal motion of the upper plane. The state of this system is described by the point of contact of the sphere with the lower plane and the orientation of the sphere in three-dimensional space. One must roll the sphere from a prescribed initial state to a prescribed terminal state so that the curve drawn by the point of contact in the plane has the minimum length. The velocity of the upper plane or, equivalently, the velocity of the centre of the sphere is the control. We consider the kinematics of the system, so can ignore the presence of the upper plane and consider how the sphere rolls over the (lower) plane without slipping or twisting. ‘No slipping’ means that the instantaneous velocity of the point of contact of the sphere to the plane is zero, and ‘no twisting’ means that the vector of angular velocity of the sphere is horizontal. Rolling a surface over other surface without twisting or slipping models the work of a robotic arm, and problems concerning such motions are of great interest in mechanics, robotics, and control theory (see, for instance, [6], [48], [97], [100], and [108]). A mathematical setting. Let $e_1$, $e_2$, $e_3$ is a fixed right-handed frame in $\mathbb{R}^3$ such that the vectors $e_1$ and $e_2$ lie in the plane $\mathbb{R}^2 \cong (\mathbb{R}^2,0) \subset \mathbb{R}^3$ on which a unit sphere $S^2$ is rolling, and $e_3$ points to the half-space containing this sphere. The frame $e_1$, $e_2$, $e_3$ is fixed at a point $O \in (\mathbb{R}^2,0)$. Let $f_1$, $f_2$, $f_3$ be a movable right-handed frame attached to the rolling sphere $S^2$. We denote the coordinates of a point in $\mathbb{R}^3$ with respect to the basis $e_1$, $e_2$, $e_3$ by $(x,y,z)$, and its coordinates with respect to the basis $f_1$, $f_2$, $f_3$ shifted to $O$ by $(X,Y,Z)$. Thus, Let $R \in \operatorname{SO}(3)$ be the matrix taking the coordinates of a point with respect to the fixed frame $e_1$, $e_2$, $e_3$ to its coordinates with respect to the movable frame $f_1$, $f_2$, $f_3$, that is,
$$
\begin{equation*}
\begin{pmatrix} X \\ Y \\ Z \end{pmatrix}= R \begin{pmatrix} x \\ y \\ z \end{pmatrix}.
\end{equation*}
\notag
$$
The state of the system ‘the sphere $S^2$ and the plane $\mathbb{R}^2$’ is described by the coordinates $(x,y)$ of the point of contact between $S^2$ and $\mathbb{R}^2$ and the rotation matrix $R$. As control we take the vector $(u_1,u_2)$ of the velocity of the centre of the sphere. The problem of the optimal rolling of the sphere over the plane is formalized as the optimal control problem
$$
\begin{equation}
Q=(x,y,R) \in G=\mathbb{R}^2 \times \operatorname{SO}(3),
\end{equation}
\tag{2.54}
$$
$$
\begin{equation}
Q(0)=Q_0=(0, 0, \operatorname{Id}), \qquad Q(t_1)=Q_1,
\end{equation}
\tag{2.56}
$$
Here and in what follows we use basis matrices of the Lie algebra $\mathfrak{so}(3)$, namely,
$$
\begin{equation}
A_1=\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix},\quad A_2=\begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix},\quad\text{and}\quad A_3=\begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}.
\end{equation}
\tag{2.58}
$$
The left-invariant sub-Riemannian problem. Problem (2.51)–(2.57) is a left- invariant sub-Riemannian problem on the Lie group $G=\mathbb{R}^2 \times \operatorname{SO}(3)$. We consider the following left-invariant frame on this group:
$$
\begin{equation*}
e_1=\frac{\partial}{\partial x}\,, \quad e_2=\frac{\partial}{\partial y}\,, \quad V_i(R)=R A_i, \quad i=1,2,3.
\end{equation*}
\notag
$$
In terms of the left-invariant fields the control system (2.51)–(2.55) assumes the following form:
$$
\begin{equation}
\dot Q=u_1 X_1 (Q)+u_2 X_2(Q), \qquad Q \in G=\mathbb{R}^2 \times \operatorname{SO}(3), \quad (u_1, u_2) \in \mathbb{R}^2.
\end{equation}
\tag{2.59}
$$
The functional (2.57) is the sub-Riemannian length functional for the left-invariant sub-Riemannian structure defined by the fields $X_1$ and $X_2$ as an orthonormal basis:
$$
\begin{equation}
\begin{gathered} \, l=\int_0^{t_1} \langle \dot Q,\dot Q\rangle^{1/2} \, dt \to \min, \\ \langle X_i,X_j\rangle=\delta_{ij}, \qquad i,j=1, 2. \nonumber \end{gathered}
\end{equation}
\tag{2.60}
$$
The existence of optimal controls. Recall that the matrix commutators $[A_i,A_j]=A_i A_j-A_j A_i$ are as follows:
$$
\begin{equation*}
[A_1, A_2]=A_3, \qquad [A_2, A_3]=A_1, \qquad [A_3, A_1]=A_2.
\end{equation*}
\notag
$$
The multiplication table in the Lie algebra $\mathfrak{g}=\mathbb{R}^2 \oplus \mathfrak{so}(3)= \operatorname{span}(e_1, e_2, V_1, V_2, V_3)$ of $G$ has the following form:
$$
\begin{equation*}
\operatorname{ad}e_i=0, \qquad [V_1, V_2]=V_3, \qquad [V_2, V_3]=V_1, \qquad [V_3, V_1]=V_2.
\end{equation*}
\notag
$$
By the equalities
$$
\begin{equation*}
[X_1, X_2]=V_3, \qquad [X_1, V_3]=- V_1, \quad\text{and}\quad [X_2, V_3]=- V_2
\end{equation*}
\notag
$$
the fields $X_1$ and $X_2$ on the right-hand side of (2.59) generate a Lie algebra $\mathfrak{g}$. By the Rashevskii–Chow theorem system (2.59) is completely controllable. By Filippov’s theorem optimal controls in the class of essentially bounded measurable controls exist in problem (2.51)–(2.57) for any $Q_0,Q_1 \in G$. 2.8.3. ExtremalsConsider the Hamiltonians
$$
\begin{equation*}
h_i(\lambda)=\langle \lambda, e_i \rangle, \qquad i=1,2,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
H_i(\lambda)=\langle \lambda, V_i \rangle, \qquad i=1,2,3,
\end{equation*}
\notag
$$
which are linear on fibres of $T^*G$. Abnormal trajectories. Constant-velocity abnormal trajectories have the form
$$
\begin{equation*}
\begin{gathered} \, x_t=u_1 t, \qquad y_t=u_2 t, \\ R_t=\exp(t(u_2 A_1-u_1 A_2)). \end{gathered}
\end{equation*}
\notag
$$
They are non-strictly abnormal. In the abnormal case the sphere rolls uniformly along a straight line. The normal Hamiltonian system. In the normal case the Hamiltonian system with Hamiltonian is expressed in coordinates as follows: As always in sub-Riemannian problems, we can limit ourselves to geodesics of unit velocity, that is, to extremal trajectories such that $H \equiv 1/2$ on them. Under this condition, in the adjoint space it is convenient to go over from the variables $(h_1,h_2,H_1,H_2,H_3)$ to new variables $(r,\alpha,\theta,c)$:
$$
\begin{equation}
h_1-H_2=\cos(\theta+\alpha), \qquad h_2+H_1=\sin (\theta+\alpha),
\end{equation}
\tag{2.67}
$$
Then the Hamiltonian system for normal extremals (2.61)–(2.65) takes the following form:
$$
\begin{equation}
\dot R =R \Omega, \qquad \Omega=\sin(\theta+\alpha) A_1-\cos(\theta+\alpha) A_2.
\end{equation}
\tag{2.73}
$$
The family of normal extremals $\lambda_t$ is parametrized by the cylinder $C$ of initial points $\lambda=\lambda_t\big|_{t=0}$:
$$
\begin{equation*}
\begin{aligned} \, C &= \biggl\{\lambda \in\mathfrak{g}^* \mid H(\lambda)=\frac{1}{2}\biggr\} \\ & \cong \{(h_1, h_2, H_1, H_2, H_3) \in \mathbb{R}^5 \mid (h_1-H_2)^2+(h_2+H_1)^2=1\} \\ & \cong \{(\theta, c, \alpha, r) \mid \theta \in S^1, \ c \in \mathbb{R}, \ \alpha \in S^1, \ r \geqslant 0\}. \end{aligned}
\end{equation*}
\notag
$$
The exponential map is defined by
$$
\begin{equation*}
\begin{gathered} \, \operatorname{Exp}(\lambda,t)=\pi \circ e^{t \vec H}(\lambda)=Q_t, \\ \operatorname{Exp}\colon N \to M, \qquad N=C \times \mathbb{R}_+=\{(\lambda,t) \mid \lambda \in C, \ t > 0\}. \end{gathered}
\end{equation*}
\notag
$$
If $r=0$, then the elastica $(x_t,y_t)$ is a straight line (for $H_3=c=0$) or a circle (for $H_3=c \ne 0$); we say that such elasticae are degenerate. If $r \ne 0$, then the elastica $(x_t,y_t)$ belongs to one of the four classes depending on the full energy $E=c^2/2-r\cos \theta$ of the pendulum (2.68), (2.69) (see § 2.6):
The symplectic foliation. On the Lie coalgebra $\mathfrak{g}^*$ we have the Casimir functions $h_1$, $h_2$, and $M=H_1^2+H_2^2+H_3^2$. The symplectic foliations if formed by
The normal Hamiltonian system has the integrals $h_1$, $h_2$, $M$, and $E=(M+h_1^2+h_2^2)/2-H$; it is integrable by elliptic functions and integrals. Geodesics of different types projecting onto Euler elasticae $(x_t,y_t)$ correspond to different types of intersection of a level surface of the Hamiltonian $\{H=\operatorname{const}\}$ with symplectic leaves. Straightening coordinates. The cylinder $C=\{\lambda \in \mathfrak{g}^* \mid H(\lambda)=1/2\}$ is stratified in accordance with types of motion of the pendulum (2.68), (2.69): where
$$
\begin{equation*}
\begin{gathered} \, C_1=\{\lambda \in C \mid E \in (-r, r), \ r > 0\}, \\ C_2=\{\lambda \in C \mid E \in (r,+\infty), \ r > 0\}, \\ C_3=\{\lambda \in C \mid E=r > 0, \ c \ne 0\}, \\ C_4=\{\lambda \in C \mid E=-r, \ r > 0\}, \\ C_5=\{\lambda \in C \mid E=r > 0, \ c=0\}, \\ C_6=\{\lambda \in C \mid r=0, \ c \ne 0\}, \end{gathered}
\end{equation*}
\notag
$$
and In the domain $\bigcup_{i=1}^3 C_i$ we can introduce coordinates $(\varphi,k,\alpha,r)$ straightening the pendulum equations (2.68), (2.69). If $\lambda=(\theta,c,\alpha,r)\in C_1$, then
$$
\begin{equation*}
\sin\frac{\theta}{2}=k\operatorname{sn}(\sqrt{r}\,\varphi,k), \quad \cos\frac{\theta}{2}=\operatorname{dn}(\sqrt{r}\,\varphi,k), \quad\text{and}\quad \frac{c}{2}=k \sqrt{r}\,\operatorname{cn}(\sqrt{r}\,\varphi, k),
\end{equation*}
\notag
$$
here $k=\sqrt{(E+r)/(2r)} \in (0, 1)$ and $\sqrt{r}\,\varphi\,\operatorname{mod}{4K} \in [0,4K]$. If $\lambda=(\theta,c,\alpha,r)\in C_2$, then
$$
\begin{equation*}
\sin\frac{\theta}{2}= \pm\operatorname{sn}\biggl(\frac{\sqrt{r}\,\varphi}{k}\,,k\biggr), \ \ \cos\frac{\theta}{2}= \operatorname{cn}\biggl(\frac{\sqrt{r}\,\varphi}{k}\,,k\biggr), \ \ \text{and}\ \ \frac{c}{2}=\pm \frac{\sqrt{r}}{k} \operatorname{dn}\biggl(\frac{\sqrt{r}\,\varphi}{k}\,,k\biggr),
\end{equation*}
\notag
$$
where $ \pm=\operatorname{sign}c$; here $k=\sqrt{2r/(E+r)} \in (0,1)$ and $\sqrt{r}\,\varphi\,\operatorname{mod}{2kK} \in [0,2kK]$. If $\lambda \in C_3$, then
$$
\begin{equation*}
\sin\frac{\theta}{2}=\pm \tanh(\sqrt{r}\,\varphi), \quad \cos\frac{\theta}{2}=\frac{1}{\cosh(\sqrt{r}\,\varphi)}\,, \quad\text{and}\quad \frac{c}{2}=\pm \frac{\sqrt{r}}{\cosh(\sqrt{r} \varphi)}\,,
\end{equation*}
\notag
$$
where $\pm=\operatorname{sign} c$; here $k=1$ and $\varphi \in (-\infty,+\infty)$. In the new variables the pendulum equations (2.68), (2.69) take the form
$$
\begin{equation*}
\dot \varphi=1, \quad \dot k=0, \quad \dot \alpha=0, \quad \dot r=0,
\end{equation*}
\notag
$$
so that $\varphi_t=\varphi+t$ and $k,\alpha,r=\operatorname{const}$. Integrating the vertical subsystem of the Pontryagin maximum principle. If $\lambda \in C_1$, then
$$
\begin{equation*}
\sin\frac{\theta_t}{2}=k\operatorname{sn}(\sqrt{r}\,\varphi_t,k), \quad \cos\frac{\theta_t}{2}=\operatorname{dn}(\sqrt{r}\,\varphi_t,k), \quad\text{and}\quad \frac{c_t}{2}=k\sqrt{r}\,\operatorname{cn}(\sqrt{r}\,\varphi_t,k).
\end{equation*}
\notag
$$
If $\lambda \in C_2$, then
$$
\begin{equation*}
\sin\frac{\theta_t}{2}= \pm \operatorname{sn}\biggl(\frac{\sqrt{r}\,\varphi_t}{k}\,,k\biggr), \quad \cos\frac{\theta_t}{2}= \operatorname{cn}\biggl(\frac{\sqrt{r}\,\varphi_t}{k}\,,k\biggr), \quad \frac{c_t}{2}=\pm \frac{\sqrt{r}}{k}\operatorname{dn} \biggl(\frac{\sqrt{r}\,\varphi_t}{k}\,,k\biggr),
\end{equation*}
\notag
$$
where $\pm=\operatorname{sign}c$. If $\lambda \in C_3$, then
$$
\begin{equation*}
\sin\frac{\theta_t}{2}= \pm \tanh(\sqrt{r}\,\varphi_t), \quad \cos\frac{\theta_t}{2}=\frac{1}{\cosh(\sqrt{r}\,\varphi_t)}\,, \quad \frac{c_t}{2}=\pm \frac{\sqrt{r}}{\cosh(\sqrt{r}\,\varphi_t)}\,,
\end{equation*}
\notag
$$
where $\pm=\operatorname{sign}c$. In the case when $\lambda \in \bigcup_{i=4}^7 C_i$ system (2.68)–(2.70) is integrated directly:
$$
\begin{equation*}
\begin{gathered} \, \theta_t \equiv 0,\quad c_t \equiv 0 \quad\text{for } \lambda \in C_4;\qquad \theta_t \equiv \pi,\quad c_t \equiv 0\quad\text{for } \lambda \in C_5; \\ \theta_t=c t+\theta,\quad c_t \equiv c \ne 0\quad\text{for } \lambda \in C_6;\qquad \theta_t \equiv \theta,\quad c_t \equiv 0\quad\text{for }\lambda \in C_7. \end{gathered}
\end{equation*}
\notag
$$
Integrating the equations for $x$ and $y$. To integrate equations (2.71), (2.72) with initial condition $x_0=y_0=0$ we use the symmetry of the problem and perform the rotation
$$
\begin{equation*}
x=\bar x\cos \alpha-\bar y \sin \alpha, \quad y=\bar x \sin \alpha+\bar y \cos \alpha.
\end{equation*}
\notag
$$
In the new variables we obtain the Cauchy problem
$$
\begin{equation}
\dot{\bar x}_t=\cos \theta_t, \quad \dot{\bar y}_t=\sin \theta_t, \qquad \bar x_0= \bar y_0=0,
\end{equation}
\tag{2.74}
$$
whose solutions can be parametrized as follows. If $\lambda \in C_1$, then
$$
\begin{equation*}
\bar{x}_t=\frac{2(\operatorname{E}(\sqrt{r}\,\varphi_t)- \operatorname{E}(\sqrt{r}\,\varphi))-\sqrt{r}\,t}{\sqrt{r}}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\bar{y}_t=\frac{2k(\operatorname{cn}(\sqrt{r}\,\varphi)- \operatorname{cn}(\sqrt{r}\,\varphi_t))}{\sqrt{r}}\,.
\end{equation*}
\notag
$$
If $\lambda \in C_2$, then
$$
\begin{equation*}
\bar{x}_t=\frac{2(\operatorname{E}(\sqrt{r}\,\varphi_t/k)- \operatorname{E}(\sqrt{r}\,\varphi/k)-(2-k^2)\sqrt{r}\,t/(2k))}{k\sqrt{r}}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\bar{y}_t=\pm\frac{2(\operatorname{dn}(\sqrt{r}\,\varphi/k)- \operatorname{dn}(\sqrt{r}\,\varphi_t/k))}{k\sqrt{r}}\,, \qquad \pm=\operatorname{sign}c.
\end{equation*}
\notag
$$
If $\lambda \in C_3$, then
$$
\begin{equation*}
\bar{x}_t=\frac{2(\tanh(\sqrt{r}\,\varphi_t)-\tanh(\sqrt{r}\,\varphi))- \sqrt{r}\,t}{\sqrt{r}}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\bar{y}_t=\pm \frac{2(1/\cosh(\sqrt{r}\,\varphi)- 1/\cosh(\sqrt{r}\,\varphi_t))}{\sqrt{r}}\,, \qquad \pm=\operatorname{sign}c.
\end{equation*}
\notag
$$
For $\lambda \in \bigcup_{i=4}^7 C_i$ equations (2.74) are integrated directly:
$$
\begin{equation*}
\begin{gathered} \, \bar{x}_t=t,\quad \bar{y}_t=0\quad\text{for } \lambda \in C_4;\qquad \bar{x}_t=-t,\quad \bar{y}_t=0\quad\text{for } \lambda \in C_5; \\ \text{and}\quad \bar{x}_t=\frac{\sin(c t+\theta)-\sin \theta}{c}\,,\quad \bar{y}_t=\frac{\cos \theta-\cos(c t+\theta)}{c}\quad\text{for } \lambda \in C_6; \\ \bar{x}_t=t \cos \theta,\quad \bar{y}_t=t \sin \theta\quad\text{for } \lambda \in C_7. \end{gathered}
\end{equation*}
\notag
$$
Integrating the equations for $R$. Assume that $M=H_1^2+H_2^2+H_3^2 > 0$. Then
$$
\begin{equation}
\begin{aligned} \, \nonumber R(t)&=\exp\bigl((\alpha-\varphi_3^0) A_3\bigr) \exp\bigl(-\varphi_2^0 A_2\bigr)\exp\bigl(\varphi_1(t) A_3\bigr) \\ &\qquad\times\exp\bigl(\varphi_2(t) A_2\bigr) \exp\bigl((\varphi_3(t)-\alpha)A_3\bigr), \end{aligned}
\end{equation}
\tag{2.75}
$$
where the angles $\varphi_i$ are defined from relations (2.76)–(2.82) for $r \ne 1$ and from (2.79)–(2.83) for $r=1$ (see below), and the angle $\varphi_1$ satisfies the initial condition $\varphi_1^0=0$. The exponentials of matrices containing $\varphi_2$ and $\varphi_3$ and involved in (2.75) are expressed in terms of $\cos \varphi_2$, $\sin \varphi_2$, $\cos \varphi_3$, and $\sin \varphi_3$. These latter are expressed in terms of the variables $c$, $\cos(\theta/2)$, and $\sin(\theta/2)$ with the help of (2.76), (2.77), (2.79), (2.80). These variables were represented above as functions of elliptic coordinates or can be used directly. For $r=1$ we have $\varphi_1(t)=\sqrt M\,t/2$. We defer the integration of equation (2.78) for $r \ne 1$ to the next subsection. In the case $M=0$ we have $r=1$, $c=0$, and $\theta=0$, so that $u_1=\cos\alpha$ and $u_2=\sin \alpha$. Therefore, $\Omega=u_2 A_1-u_1 A_2 \equiv \operatorname{const}$ and $R(t)=e^{t\Omega}$. Integrating the equations for $\varphi_1$. Along normal geodesics, for $r \ne 1$ the angles $\varphi_i$ satisfy the equalities
$$
\begin{equation}
\cos \varphi_2 =\frac{c}{\sqrt{M}}\,, \qquad \sin \varphi_2 =\pm \frac{\sqrt{M-c^2}}{\sqrt{M}}\,,
\end{equation}
\tag{2.76}
$$
$$
\begin{equation}
\cos \varphi_3 =\mp\frac{\sin \theta}{\sqrt{M-c^2}}\,, \qquad \sin \varphi_3 =\pm\frac{r-\cos \theta}{\sqrt{M-c^2}}\,,
\end{equation}
\tag{2.77}
$$
$$
\begin{equation}
\dot\varphi_1=\frac{\sqrt{M}(1-r \cos \theta)}{M-c^2}\,,
\end{equation}
\tag{2.78}
$$
while for $r=1$ they satisfy
$$
\begin{equation}
\cos \varphi_2 =\frac{c}{\sqrt{M}}\,, \qquad \sin \varphi_2 =\pm \frac{2 \sin(\theta/2)}{\sqrt{M}}\,,
\end{equation}
\tag{2.79}
$$
$$
\begin{equation}
\cos \varphi_3 =\mp \cos\frac{\theta}{2}\,, \qquad \sin \varphi_3 =\pm \sin\frac{\theta}{2}\,,
\end{equation}
\tag{2.80}
$$
We introduce an elliptic integral of the third kind in the following form:
$$
\begin{equation}
\Pi(n, u, k)=\int_0^u \frac{dt}{(1-n \sin^2 t)\sqrt{1-k^2 \sin^2 t}}=\int_0^{F(u,k)} \frac{dv}{1-n \operatorname{sn}^2 v}\,.
\end{equation}
\tag{2.82}
$$
Let $r \ne 1$. If $\lambda_1 \in C_1$, then
$$
\begin{equation}
\varphi_1(t)=\frac{\sqrt{M}}{2}\,t+\frac{\sqrt{M}\,(1+r)}{2\,\sqrt{r}\,(1-r)} \bigl[\Pi(l,\operatorname{am}(\sqrt{r}\,(\varphi+t)),k)- \Pi(l,\operatorname{am}(\sqrt{r}\,\varphi),k)\bigr],
\end{equation}
\tag{2.83}
$$
where $l=-4 k^2 r/(1-r)^2$. If $\lambda_1 \in C_2$, then
$$
\begin{equation*}
\varphi_1(t)=\frac{\sqrt{M}}{2}\,t+\frac{\sqrt{M}\,k(1+r)}{2\,\sqrt{r}\,(1-r)}\, \biggl[\Pi\biggl(l,\operatorname{am} \biggl(\frac{\sqrt{r}\,(\varphi+t)}{k}\biggr),k\biggr)- \Pi\biggl(l,\operatorname{am} \biggl(\frac{\sqrt{r}\,\varphi}{k}\biggr),k\biggr)\biggr],
\end{equation*}
\notag
$$
where $l=-4r/(1-r)^2$. If $\lambda_1 \in C_3$, then
$$
\begin{equation*}
\varphi_1(t)=\frac{\sqrt{M}}{2}\,t+\frac{\sqrt{M}\,k(1-r^2)}{8 r^{3/2}} \bigl[I(\sqrt{r}\,(\varphi+t),a)-I(\sqrt{r}\,\varphi,a)\bigr],
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
I(v,a)=\int_0^v\frac{dt}{a^2+\tanh^2t}= \frac{a t-\arctan a+\arctan (e^t(a^2\cosh t+\sinh t)/a)}{a+a^3}\,,
\end{equation*}
\notag
$$
where $ a=(1-r)/(2\,\sqrt{r}\,)$. If $\lambda_1 \in C_6$, then $\varphi_1(t)=\sqrt{1+c^2}\, t$. If $\lambda \in C_4 \cup C_5 \cup C_7$, then
$$
\begin{equation*}
\theta_t \equiv \operatorname{const}=\theta,\quad \Omega=\sin(\alpha+\theta)A_1-\cos(\alpha+\theta) A_2 \equiv \operatorname{const},\quad\text{and}\quad R(t)=e^{t\Omega}.
\end{equation*}
\notag
$$
The control system in terms of quaternions. To describe the orientation of the rolling sphere it is convenient to use quaternions along with the rotation matrix $R$. Let
$$
\begin{equation*}
\mathbb{H}=\{q=q_0+iq_1+j q_2+k q_3 \mid q_0,\dots,q_3 \in \mathbb{R}\}
\end{equation*}
\notag
$$
be the algebra of quaternions, let $S^3=\{q \in \mathbb{H} \mid |q|^2=q_0^2+q_1^2+q_2^2+q_3^2=1\}$ be the unit sphere, and be the subspace of purely imaginary quaternions. A quaternion $q \in S^3$ defines a rotation of the Euclidean space $I$:
$$
\begin{equation*}
q \in S^3 \quad\Longrightarrow \quad R_q(a)=q a q^{-1}, \quad a \in I, \quad R_q \in \operatorname{SO}(3)\cong \operatorname{SO}(I).
\end{equation*}
\notag
$$
The correspondence between the quaternions $q=q_0+i q_1+j q_2+k q_3 \in S^3$ and the matrices $R \in \operatorname{SO}(3)$ looks as follows:
$$
\begin{equation}
R=\begin{pmatrix} q_0^2+q_1^2-q_2^2-q_3^2 & 2 q_1 q_2-2 q_0 q_3& 2 q_0 q_2 + 2 q_1 q_3 \\ 2 q_1 q_2+2 q_0 q_3 &q_0^2-q_1^2+q_2^2-q_3^2 & -2 q_0q_1+2 q_2 q_3 \\ -2 q_0 q_2+2 q_1 q_3 & 2 q_0 q_1+2 q_2 q_3 & q_0^2-q_1^2 - q_2^2+q_3^2 \end{pmatrix}.
\end{equation}
\tag{2.84}
$$
In terms of quaternions the control system (2.47) takes the following form:
$$
\begin{equation}
\begin{cases} \dot{q}_0=(q_2 u_1-q_1 u_2)/2, \\ \dot{q}_1=(q_3 u_1+q_0 u_2)/2, \\ \dot{q}_2=(-q_0 u_1+q_3 u_2)/2, \\ \dot{q}_3=(-q_1 u_1-q_2 u_2)/2, \end{cases}\qquad q=q_0+i q_1+j q_2+k q_3 \in S^3, \quad (u_1,u_2) \in \mathbb{R}^2.
\end{equation}
\tag{2.85}
$$
The control system on $\mathbb{R}^2\times\operatorname{SO}(3)$ (2.46), (2.47) with initial condition $g(0)=\operatorname{Id}$ lifts to $\mathbb{R}^2\times S^3$ as a system (2.46), (2.85) with initial conditions $(x,y)(0)=(0,0)$ and $q(0)=1$. 2.8.4. SymmetriesSymmetries of the family of extremal trajectories. The rotations of elasticae $(x_s,y_s)$ around the origin in the $(x,y)$-plane generate a one- parameter group of symmetries of trajectories of the Hamiltonian system (2.68)–(2.73) where the rotation $\Phi^{\beta}$ is defined as follows:
$$
\begin{equation}
\Phi^{\beta}\colon\{\lambda_s \mid s \in [0,t]\}\to \{\lambda_s^{\beta} \mid s \in [0,t]\},
\end{equation}
\tag{2.86}
$$
where
$$
\begin{equation}
\lambda_s=(\theta_s, c_s, \alpha, r, Q_s), \qquad Q_s=(x_s, y_s, R_s),
\end{equation}
\tag{2.87}
$$
$$
\begin{equation}
\lambda_s^{\beta}= (\theta_s^{\beta},c_s^{\beta},\alpha^{\beta},r,Q_s^{\beta}),\qquad Q_s^{\beta}=(x_s^{\beta},y_s^{\beta},R_s^{\beta}),
\end{equation}
\tag{2.88}
$$
$$
\begin{equation}
\theta_s^{\beta}=\theta_s, \qquad c_s^{\beta}=c_s, \qquad \alpha^{\beta}=\alpha+\beta,
\end{equation}
\tag{2.89}
$$
$$
\begin{equation}
\begin{pmatrix} x_s^{\beta} \\ y_s^{\beta}\end{pmatrix}= \begin{pmatrix} \cos \beta &-\sin \beta \\ \sin \beta & \cos \beta \end{pmatrix} \begin{pmatrix} x_s \\ y_s\end{pmatrix},
\end{equation}
\tag{2.90}
$$
Proposition 2.8. If $\{\lambda_s \mid s \in [0,t]\}$ is a trajectory of system (2.68)–(2.73), then for each $\beta \in S^1$ the curve $\{\lambda_s^{\beta} \mid s \in [0,t]\}$ is a trajectory of this system too. The reflections of trajectories $(\theta_s,c_s)$ of the pendulum (2.68), (2.69) in the $\theta$- and $c$-coordinate axes and in the origin extend to discrete symmetries $\varepsilon^1$, $\varepsilon^2$, and $\varepsilon^3$ of the family of trajectories of system (2.68)–(2.73):
$$
\begin{equation*}
\begin{gathered} \, \varepsilon^i\colon\{\lambda_s \mid s \in [0,t]\}\to \{\lambda_s^{i} \mid s \in [0,t]\}, \qquad i=1,2,3, \\ \begin{alignedat}{2} \lambda_s&=(\theta_s, c_s, \alpha, r, Q_s), &\qquad Q_s&=(x_s, y_s, R_s), \\ \lambda_s^{i}&=(\theta_s^{i}, c_s^{i}, \alpha^{i}, r, Q_s^{i}), &\qquad Q_s^{i}&=(x_s^{i}, y_s^{i}, R_s^{i}). \end{alignedat} \end{gathered}
\end{equation*}
\notag
$$
The reflection of trajectories $(\theta_s,c_s)$ of the pendulum (2.68), (2.69) in the $\theta$-axis corresponds to a discrete symmetry $\varepsilon^1$ of the family of extremal trajectories:
$$
\begin{equation*}
\begin{gathered} \, \theta_s^{1}=\theta_{t-s}, \qquad c_s^{1}=-c_{t-s}, \qquad \alpha^{1}=\alpha+\pi, \\ x_s^1=x_{t-s}-x_t, \qquad y_s^1=y_{t-s}-y_t, \\ R_s^{1}=(R_t)^{-1} R_{t-s}, \qquad \Omega_s^{1}=-\Omega_{t-s}. \end{gathered}
\end{equation*}
\notag
$$
The reflection of trajectories $\theta_s,c_s)$ of the pendulum in the $c$-axis generates a symmetry $\varepsilon^2$ of extremal trajectories:
$$
\begin{equation*}
\begin{gathered} \, \theta_s^{2}=-\theta_{t-s}, \qquad c_s^{2}=c_{t-s}, \qquad \alpha^{2}=\pi-\alpha, \\ x_s^2=x_{t-s}-x_t, \qquad y_s^2=y_t-y_{t-s}, \\ R_s^{2}=I_2 (R_t)^{-1} R_{t-s} I_2, \qquad \Omega_s^{2}=- I_2 \Omega_{t-s} I_2, \\ I_2=I_2^{-1}=e^{\pi A_2}=\begin{pmatrix} - 1 & 0 & \hphantom{-}0 \\ \hphantom{-}0 & 1 & \hphantom{-}0 \\ \hphantom{-}0 & 0 & -1 \end{pmatrix}. \end{gathered}
\end{equation*}
\notag
$$
The reflection of trajectories $(\theta_s,c_s)$ of the pendulum in the origin $(\theta,c)=(0,0)$ extends to a symmetry $\varepsilon^3$ of extremal trajectories:
$$
\begin{equation*}
\begin{gathered} \, \theta_s^{3}=-\theta_{s}, \qquad c_s^{3}=-c_{s}, \qquad \alpha^{3}=- \alpha, \\ x_s^3=x_{s}, \qquad y_s^3=- y_{s}, \\ R_s^{3}=I_2 R_{s} I_2, \qquad \Omega_s^{3}=I_2 \Omega_{s} I_2. \end{gathered}
\end{equation*}
\notag
$$
Proposition 2.9. If $\{\lambda_s \mid s \in [0,t]\}$ is a trajectory of system (2.68)–(2.73), then the curves $\{\lambda_s^{i} \mid s \in [0,t]\}$, $i=1,2,3$, are also trajectories of these system. Symmetries of the exponential map. The actions of rotations $\Phi^\beta$ and reflections $\varepsilon^i$ on the source space and target space of the exponential map are defined so as to commute with the action of the exponential map. Rotations $\Phi^\beta\colon\lambda\to\lambda^{\beta}$ (2.86)–(2.91) are symmetries of the Hamiltonian system, so their actions on $T^*G$ decompose in a natural way into a direct sum of actions on $N=\mathfrak{g}^* \times \mathbb{R}_+$ (on points $(\lambda,t)$, where $\lambda$ is the initial point of the extremal) and on $G$ (on the terminal points $Q_t$ of the corresponding extremal trajectories):
$$
\begin{equation*}
\begin{gathered} \, \Phi^\beta\colon N \to N, \qquad (\lambda,t) \mapsto (\lambda^{\beta},t), \\ \lambda=(\theta,c,\alpha,r), \qquad \lambda^{\beta}=(\theta,c,\alpha^{\beta},r),\qquad \alpha^{\beta}=\alpha+\beta, \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{gathered} \, \Phi^\beta\colon G\to G, \qquad Q \mapsto Q^{\beta}, \\ Q=(x, y, R), \qquad Q^{\beta}=(x^{\beta}, y^{\beta}, R^{\beta}), \\ \begin{pmatrix} x^{\beta} \\ y^{\beta}\end{pmatrix}=\begin{pmatrix} \cos \beta &-\sin \beta \\ \sin \beta & \hphantom{-}\cos \beta \end{pmatrix} \begin{pmatrix} x \\ y\end{pmatrix},\qquad R^{\beta}=e^{\beta A_3} R e^{-\beta A_3}. \end{gathered}
\end{equation*}
\notag
$$
The action of the reflections $\varepsilon^i$ on $N$ is defined by their restrictions to the vertical components of extremal trajectories at the initial moment of time $s=0$:
$$
\begin{equation*}
\begin{gathered} \, \varepsilon^i\colon N\to N, \quad (\lambda, t) \mapsto (\lambda^{i}, t), \qquad i=1, 2, 3, \\ \lambda=(\theta, c, \alpha, r), \qquad \lambda^{i}=(\theta^i, c^i, \alpha^{i}, r), \end{gathered}
\end{equation*}
\notag
$$
where $\lambda=\lambda_s\big|_{s=0}$ and $\lambda^i=\lambda_s^i\big|_{s=0}$. The explicit expressions for the actions of the $\varepsilon^i$ on $N$ are as follows:
$$
\begin{equation*}
\begin{aligned} \, \varepsilon^1\colon(\theta,c,\alpha,r,t)\to(\theta^1,c^1,\alpha^1,r,t)&= (\theta_t,-c_t,\alpha+\pi,r,t), \\ \varepsilon^2\colon(\theta,c,\alpha,r,t)\to(\theta^2,c^2,\alpha^2,r,t)&= (-\theta_t, c_t, \pi-\alpha, r, t), \\ \varepsilon^3\colon(\theta,c,\alpha,r,t)\to(\theta^3,c^3,\alpha^3,r,t)&= (-\theta, -c, -\alpha, r, t). \end{aligned}
\end{equation*}
\notag
$$
The action of reflections on $G$ is determined by their action on the extremal trajectories at the terminal moment of time $s=t$:
$$
\begin{equation*}
\begin{gathered} \, \varepsilon^i\colon G\to G, \quad Q \mapsto Q^{i}, \qquad i=1,2,3, \\ Q=(x,y,R), \qquad Q^i=(x^i,y^i,R^i), \end{gathered}
\end{equation*}
\notag
$$
where $Q=Q_s\big|_{s=t}$ and $Q^i=Q_s^i\big|_{s=t}$. The explicit formulae are
$$
\begin{equation*}
\begin{aligned} \, \varepsilon^1\colon(x, y, R)\to(x^1, y^1, R^1)&=(-x, -y, (R)^{-1}), \\ \varepsilon^2\colon(x, y, R)\to(x^2, y^2, R^2)&=(-x, y, I_2 (R)^{-1}I_2), \\ \varepsilon^3\colon(x, y, R)\to(x^3, y^3, R^3)&=(x, -y, I_2 R I_2). \end{aligned}
\end{equation*}
\notag
$$
Thus we have defined the actions of rotations and reflections on the target space of the exponential map:
$$
\begin{equation}
\Phi^\beta,\varepsilon^i \colon N \to N, \qquad (\lambda,t) \mapsto (\lambda^{\beta},t),(\lambda^i,t),
\end{equation}
\tag{2.92}
$$
$$
\begin{equation}
\Phi^\beta,\varepsilon^i \colon G \to G, \qquad Q \mapsto Q^{\beta},Q^i.
\end{equation}
\tag{2.93}
$$
It is essential that the image $Q^i=\varepsilon^i(Q)$ depends only on the preimage $Q$, rather than on the moment of time $t$. Consider the group of symmetries of the exponential map generated by the rotations and reflections:
$$
\begin{equation*}
\operatorname{Sym}= \langle \Phi^\beta,\varepsilon^1,\varepsilon^2,\varepsilon^3 \rangle.
\end{equation*}
\notag
$$
The multiplication table in this group is as follows:
Hence we obtain an explicit description of the symmetry group of the exponential map:
$$
\begin{equation*}
\operatorname{Sym}=\{\Phi^\beta,\Phi^\beta \circ \varepsilon^i \mid \beta \in S^1, \ i=1,2,3\} \cong \operatorname{SO}(2)\times (\mathbb{Z}_2\times\mathbb{Z}_2).
\end{equation*}
\notag
$$
The Maxwell set corresponding to the group $\langle \varepsilon^i,\Phi^\beta\rangle$, $i=1,2,3$, is defined by
$$
\begin{equation*}
\begin{aligned} \, \operatorname{MAX}^i=\bigl\{(\lambda,t)\in N\mid \exists\beta \in S^1&\colon (\tilde\lambda,t)=\varepsilon^i \circ \Phi^\beta(\lambda,t), \\ &\quad\operatorname{Exp}(\lambda, s) \not\equiv \operatorname{Exp}(\tilde\lambda, s), \ \operatorname{Exp}(\lambda,t)= \operatorname{Exp}(\tilde \lambda,t)\bigr\}. \end{aligned}
\end{equation*}
\notag
$$
2.8.5. Conditions for optimalityTheorem 2.36. Let $t > 0$, and let $Q_s=(x_s, y_s, R_s)=\operatorname{Exp}(\lambda,s)$ be an extremal trajectory such that Then $(\lambda,t) \in \operatorname{MAX}^1$, so that for no $t_1 > t$ is the trajectory $Q_s$, $s \in [0,t_1]$, optimal. Theorem 2.37. Let $t > 0$, and let $Q_s=(x_s,y_s,R_s)=\operatorname{Exp}(\lambda,s)$ be an extremal trajectory such that Then $(\lambda,t) \in \operatorname{MAX}^2$, so that for no $t_1 > t$ is the trajectory $Q_s$, $s \in [0, t_1]$, optimal. Theorem 2.38. Let $t > 0$, and let $Q_s=(x_s, y_s, R_s)=\operatorname{Exp}(\lambda,s)$ be an optimal trajectory such that Then $(\lambda,t) \in \operatorname{MAX}^3$, so that for no $t_1 > t$ is the trajectory $Q_s$, $s \in [0,t_1]$, optimal. Remark 2.2. Bearing in mind that for each quaternion $q=q_0+iq_1+jq_2+kq_3 \in S^3$ the corresponding motion $R_q\colon \mathbb{R}^3 \to \mathbb{R}^3$ is a rotation about the vector $(q_1,q_2,q_3) \in\mathbb{R}^3$, we can give the following geometric interpretation of conditions (1) in Theorems 2.36–2.38. 1. Condition (1) in Theorem 2.36 means that the rotation $R_t$ of the sphere is a rotation about some horizontal axis. 2. Condition (1) in Theorem 2.37 means that $R_t$ is a rotation about some axis orthogonal to the displacement vector of the point of contact of the sphere and the plane $(x_t,y_t,0)$. 3. Condition (1) in Theorem 2.38 means that $R_t$ is a rotation about the horizontal axis orthogonal to the vector $(x_t,y_t,0)$, or the rotation through $\pi$ about an axis lying in the vertical plane containing $(x_t,y_t,0)$. 2.8.6. Bibliographic commentsSubsection 2.8.1 is based on [29], [56], [83], and [86], § 2.8.2 is based on [132], § 2.8.3 on [114] and [132], and § 2.8.4 on [132]. The problem of rolling a ball over a plane without twisting or slipping was also considered in [2] and [88]. 2.9. The sub-Riemannian problem on the Engel group2.9.1. The problem statementThe geometric setting. Fix two points $a_0,a_1 \in \mathbb{R}^2$ on the plane which are connected by a curve $\gamma_0 \subset \mathbb{R}^2$. Also fix $S \in \mathbb{R}$ and a line $L \subset \mathbb{R}^2$. The problem consists in joining $a_0$ with $a_1$ by a shortest curve $\gamma \subset \mathbb{R}^2$ such that $\gamma_0$ and $\gamma$ bound a plane domain with algebraic area $S$ whose centre of mass lies on $L$. Thus this is a generalization (a sophistication) of the Dido problem (see [143] and [157]). The optimal control problem. The above geometric problem can also be formulated as an optimal control problem:
$$
\begin{equation}
\dot{g}=u_1 X_1(g)+u_2 X_2(g), \quad g=(x, y, z, v) \in \mathbb{R}^4,
\end{equation}
\tag{2.94}
$$
$$
\begin{equation}
X_1=\frac{\partial}{\partial x}- \frac{y}{2}\,\frac{\partial}{\partial z}\,, \quad X_2=\frac{\partial}{\partial y}+ \frac{x}{2}\,\frac{\partial}{\partial z}+ \frac{x^2+y^2}{2}\,\frac{\partial}{\partial v}\,.
\end{equation}
\tag{2.97}
$$
This is a sub-Riemannian problem for the sub-Riemannian structure on $\mathbb{R}^4$ defined by the vector fields $X_1$ and $X_2$ treated as an orthonormal frame. The Engel algebra and Engel group. The Engel algebra is the Lie algebra $\mathfrak{g}$, with basis $(X_1,\dots,X_4)$ such that the non-trivial commutators of its elements are (see Fig. 34).The Engel algebra is a nilpotent Lie algebra with grading:
$$
\begin{equation*}
\mathfrak{g}=\mathfrak{g}^{(1)} \oplus\mathfrak{g}^{(2)} \oplus \mathfrak{g}^{(3)},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\mathfrak{g}^{(1)}=\operatorname{span}(X_1,X_2),\quad \mathfrak{g}^{(2)}=\mathbb{R}X_3,\quad \mathfrak{g}^{(3)}=\mathbb{R} X_4,\qquad [\mathfrak{g}^{(1)},\mathfrak{g}^{(i)}]=\mathfrak{g}^{(i+1)},
\end{equation*}
\notag
$$
and $\mathfrak{g}^{(4)}=\{0\}$, so it is a Carnot algebra. The corresponding connected and simply connected Lie group $G$ is called the Engel group. The Engel group has a linear representation as
$$
\begin{equation*}
\left\{\begin{pmatrix} 1 & b & c & d \\ 0 & 1 & a & a^2/2\\ 0 & 0 & 1 & a \\ 0 & 0 & 0 & 1 \end{pmatrix}\,\Bigg|\, a,b,c,d \in \mathbb{R}\right\}.
\end{equation*}
\notag
$$
In $\mathbb{R}^4_{x,y,z,v}$ we introduce multiplication by
$$
\begin{equation*}
\begin{aligned} \, \begin{pmatrix} x_1\\ y_1\\ z_1\\ v_1 \end{pmatrix} \cdot \begin{pmatrix} x_2\\ y_2\\ z_2\\ v_2 \end{pmatrix}=\begin{pmatrix} x_1+x_2\\ y_1+y_2\\ z_1+z_2+\dfrac{x_1 y_2-x_2 y_1}{2}\\ v_1+v_2+\dfrac{y_1y_2(y_1+y_2)}{2}+x_1z_2+\dfrac{x_1y_2(x_1+x_2)}{2} \end{pmatrix}, \end{aligned}
\end{equation*}
\notag
$$
which makes of it the Engel group: $G \cong \mathbb{R}^4_{x,y,z,v}$ so that the fields (2.97) become left-invariant fields on this group. Thus, (2.94)–(2.96) is a left-invariant sub- Riemannian problem on the Engel group. Hence we can assume that the reference point in (2.95) is the identity element of the Engel group: $g_0=\operatorname{Id}=(0,0,0,0)$. All completely non-holonomic left-invariant sub-Riemannian problems of rank 2 on the Engel group are transformed into one another by isomorphisms of this group [124]. Special features of the problem. The sub-Riemannian problem on the Engel group is a simplest left-invariant sub-Riemannian problem, which has the following properties:
This problem presents a nilpotent approximation to each sub-Riemannian problem of Engel type (that is, with growth vector $(2,3,4)$: see [45] and [143]) and, in particular, to a mobile robot with trailer. 2.9.2. Symmetries of the distribution and sub-Riemannian structureTheorem 2.39. The Lie algebra of infinitesimal symmetries of the distribution $\operatorname{span}(X_1,X_2)$ on the Engel group is parametrized by the smooth functions on this group that are constant along the field $X_2$. Theorem 2.40. The Lie algebra of infinitesimal symmetries of the nilpotent sub- Riemannian structure on the Engel group is isomorphic to the Engel algebra of right-invariant vector field on this group. 2.9.3. GeodesicsOptimal trajectories exist in problem (2.94)–(2.96) as follows from the Rashevskii–Chow and Filippov theorems. The Pontryagin maximum principle. From the problem of the minimization of length (2.96) we go over to the equivalent problem of the minimization of energy
$$
\begin{equation}
J=\frac{1}{2}\int_0^{t_1} (u_1^2+u_2^2)\,dt \to \min\!.
\end{equation}
\tag{2.98}
$$
Consider the Hamiltonians $h_i(\lambda)=\langle\lambda,X_i\rangle$, $i=1,\dots,4$, which are constant on fibres of $T^*G$. Then the Pontryagin maximum principle for problem (2.94), (2.95), (2.98) assumes the following form:
$$
\begin{equation*}
\begin{gathered} \, \dot h_1=-u_2 h_3,\quad \dot h_2=u_1 h_3,\quad \dot h_3=u_1 h_4,\quad \dot h_4=0, \\ \dot{g}=u_1 X_1+u_2 X_2, \\ u_1 h_1+u_2 h_2+\frac{\nu}{2}(u_1^2+u_2^2) \to \max_{(u_1,u_2) \in \mathbb{R}^2}, \\ \nu \leqslant 0,\quad (h_1,\dots,h_4,\nu) \ne 0. \end{gathered}
\end{equation*}
\notag
$$
Abnormal extremals. Abnormal extremals of constant velocity can be parametrized as follows:
$$
\begin{equation}
\begin{gathered} \, h_1=h_2=h_3=0, \quad h_4 \equiv \operatorname{const} \ne 0, \nonumber \\ u_1\equiv 0, \quad u_2\equiv \pm 1, \nonumber \\ x=z \equiv 0, \quad y=\pm t, \quad v=\pm \frac{t^3}{6}\,. \end{gathered}
\end{equation}
\tag{2.99}
$$
Abnormal trajectories (2.99) are one-parameter subgroups $g(t)=e^{\pm tX_2}$. They project onto the $(x,y)$-plane as straight lines, so they are sub-Riemannian length minimizers. The abnormal set is a smooth one-dimensional manifold diffeomorphic to a line:
$$
\begin{equation*}
\operatorname{Abn}=\biggl\{g \in G \Bigm| x=z=v-\frac{y^3}{6}=0\biggr\}.
\end{equation*}
\notag
$$
Normal extremals. Normal extremals are trajectories of the normal Hamiltonian system
$$
\begin{equation}
\begin{aligned} \, \dot{\lambda}=\vec{H}(\lambda), \quad \lambda \in T^*G, \end{aligned}
\end{equation}
\tag{2.100}
$$
with Hamiltonian $H=(h_1^2+h_2^2)/2$. On the level surface $\{H=1/2\}$ we introduce variables $(\theta,c,\alpha)$:
$$
\begin{equation*}
h_1=-\sin\theta, \quad h_2=\cos\theta, \quad h_3=c, \quad h_4=\alpha.
\end{equation*}
\notag
$$
Then the Hamiltonian system (2.100) takes the form
$$
\begin{equation}
\dot{\theta}=c, \quad \dot c=-\alpha\sin\theta, \quad \dot{\alpha}=0,
\end{equation}
\tag{2.101}
$$
$$
\begin{equation}
\dot{g}=-\sin \theta\cdot X_1+\cos\theta\cdot X_2.
\end{equation}
\tag{2.102}
$$
The vertical subsystem (2.101) is the equation of a pendulum in a gravity field with gravitational acceleration $g=\alpha l$, where $l$ is the length of the pendulum. Thus, for $\alpha> 0$ (for $\alpha < 0$) the gravity force is directed upwards (downwards, respectively) relative to the axis from which we measure $\theta$, while for $\alpha=0$ the pendulum moves in zero gravity. The projections of normal extremals onto the $(x,y)$-plane are Euler elasticae (see § 2.6). Abnormal length minimizers satisfy the normal Hamiltonian system (2.101), (2.102) for $\theta=\pi+2\pi n$ and $c=0$, so they are non-strictly abnormal. The symplectic foliation and Casimir functions There exist two independent Casimir functions on the Lie coalgebra $\mathfrak{g}^*$:
$$
\begin{equation*}
\begin{aligned} \, h_4 \quad\text{and}\quad E=\frac{h_3^2}{2}-h_2 h_4, \end{aligned}
\end{equation*}
\notag
$$
where $E$ is the full energy of the pendulum (2.101). The symplectic foliation $\mathfrak{g}^*$ consists of Symplectic leaves are two- and zero-dimensional, so the vertical subsystem (2.101) is Liouville integrable. The phase portrait of the Hamiltonian system (2.100) on the cylinder $C\cap \{h_4=\operatorname{const}\}$, where $C=\mathfrak{g}^* \cap \{H=1/2\}$, is obtained as the intersections of this cylinder with level surfaces of the energy $E$.A parametrization of normal geodesics. The family of normal extremals on the level surface $\{H=1/2\}$ is parametrized by their initial points on the cylinder $C$. Consider the stratification of $C$ by the submanifolds corresponding to different types of trajectories of the pendulum (2.101): where
$$
\begin{equation*}
\begin{gathered} \, C_1=\{\lambda \in C \mid \alpha \ne 0, \ E\in(-|\alpha|,|\alpha|)\}, \\ C_2=\{\lambda \in C \mid \alpha \ne 0, \ E\in(|\alpha|,+\infty)\}, \\ C_3=\{\lambda \in C \mid \alpha \ne 0, \ E=|\alpha|, c \ne 0 \}, \\ C_4=\{\lambda \in C \mid \alpha \ne 0, \ E=-|\alpha|\}, \\ C_5=\{\lambda \in C \mid \alpha \ne 0, \ E=|\alpha|, c=0\}, \\ C_6=\{\lambda \in C \mid \alpha=0, \ c \ne 0\}, \end{gathered}
\end{equation*}
\notag
$$
and Next, the sets $C_i$, $i=1,\dots,5$, are partitioned into subsets depending on the sign of $\alpha$:
$$
\begin{equation*}
C_i^+=C_i \cap \{\alpha>0\}, \quad C_i^-=C_i \cap \{\alpha<0\}, \quad i\in\{1,\dots,5\}.
\end{equation*}
\notag
$$
Moreover, the subsets $C_6$, $C_2^{\pm}$, and $C_3^{\pm}$ fall into connected components depending on the sign of $c$:
$$
\begin{equation*}
\begin{alignedat}{2} C_{6+}&=C_6 \cap \{c>0\}, &\quad C_{6-}&=C_6 \cap \{c<0\}, \\ C_{i+}^{\pm}&=C_i^{\pm} \cap \{c>0\}, &\quad C_{i-}^{\pm}&=C_i^{\pm}\cap \{c<0\}, \qquad i\in\{2,3\}. \end{alignedat}
\end{equation*}
\notag
$$
For the normalization of normal geodesics, on the strata $C_1$, $C_2$, and $C_3$ we introduce elliptic coordinates $(\varphi,k,\alpha)$ such that the pendulum equation (2.101) straightens out in these coordinates. In the domain $C_1^+$ we set
$$
\begin{equation*}
\begin{gathered} \, k=\sqrt{\frac{E+\alpha}{2 \alpha}}= \sqrt{\frac{c^2}{4\alpha}+\sin^2 \frac{\theta}{2}}\in (0,1), \\ \sin\frac{\theta}{2}=k\operatorname{sn}(\sqrt{\alpha}\,\varphi),\qquad \cos\frac{\theta}{2}=\operatorname{dn}(\sqrt{\alpha}\,\varphi), \\ \frac{c}{2}=k\sqrt{\alpha}\,\operatorname{cn}(\sqrt{\alpha}\,\varphi),\qquad \varphi \in [0,4K]. \end{gathered}
\end{equation*}
\notag
$$
In the domain $C_2^+$ we set
$$
\begin{equation*}
\begin{gathered} \, k=\sqrt{\frac{2\alpha}{E+\alpha}}=\frac{1}{\sqrt{c^2/(4\alpha)+ \sin^2(\theta/2)}}\in (0,1), \\ \sin\frac{\theta}{2}=\operatorname{sign} c\cdot \operatorname{sn}\biggl(\frac{\sqrt{\alpha}\,\varphi}{k}\biggr),\qquad \cos\frac{\theta}{2}= \operatorname{cn}\biggl(\frac{\sqrt{\alpha}\,\varphi}{k}\biggr), \\ \frac{c}{2}=\operatorname{sign}c \cdot \frac{\sqrt{\alpha}}{k} \operatorname{dn}\biggl(\frac{\sqrt{\alpha}\,\varphi}{k}\biggr), \qquad \varphi \in [0,2kK], \\ \psi=\frac{\varphi}{k}\,. \end{gathered}
\end{equation*}
\notag
$$
On the set $C_3^+$,
$$
\begin{equation*}
\begin{gathered} \, k=1,\\ \sin\frac{\theta}{2}=\operatorname{sign}c\cdot \tanh(\sqrt{\alpha}\,\varphi),\qquad \cos\frac{\theta}{2}=\frac{1}{\cosh(\sqrt{\alpha}\,\varphi)}\,, \\ \frac{c}{2}=\operatorname{sign} c \cdot \frac{\sqrt{\alpha}}{\cosh(\sqrt{\alpha}\,\varphi)}\,, \qquad \varphi \in (-\infty,+\infty). \end{gathered}
\end{equation*}
\notag
$$
On $C_1^-$, $C_2^-$, and $C_3^-$ we define the new coordinates as follows:
$$
\begin{equation}
\varphi(\theta,c,\alpha)=\varphi(\theta-\pi,c,-\alpha),
\end{equation}
\tag{2.103}
$$
In the new variables the vertical subsystem (2.101) assumes the following form:
$$
\begin{equation*}
\dot{\varphi}=1, \qquad \dot{k}=0, \qquad \dot{\alpha}=0,
\end{equation*}
\notag
$$
so it has the solutions
$$
\begin{equation}
\varphi_t=\varphi+t, \qquad k=\operatorname{const}, \qquad \alpha=\operatorname{const}.
\end{equation}
\tag{2.105}
$$
The problem is invariant under left shifts on the Engel group, under the dilations
$$
\begin{equation}
\delta_s\colon (t,x,y,z,v) \mapsto (e^s t,e^s x,e^s y,e^{2s}z,e^{3s}v),
\end{equation}
\tag{2.106}
$$
$$
\begin{equation}
(\theta,c,\alpha) \mapsto (\theta,e^{-s}c,e^{-2s}\alpha), \quad (\varphi,k,\alpha) \mapsto (e^s\varphi,k,e^{-2s}\alpha)
\end{equation}
\tag{2.107}
$$
and under the reflections
$$
\begin{equation*}
\begin{gathered} \, (t,x,y,z,v) \mapsto (t,-x,-y,z,-v), \\ (\theta,c,\alpha) \mapsto (\theta-\pi,c,-\alpha), \quad (\varphi,k,\alpha) \mapsto (\varphi,k,-\alpha). \end{gathered}
\end{equation*}
\notag
$$
Dilations define the flow of the vector field
$$
\begin{equation*}
Y=x\,\frac{\partial}{\partial x}+y\,\frac{\partial}{\partial y}+ 2z\,\frac{\partial}{\partial z}+3v\,\frac{\partial}{\partial v}
\end{equation*}
\notag
$$
on the Engel group. For $\lambda=(\varphi,k,\alpha) \in \bigcup_{i=1}^3 C_i$, and $\alpha=1$, geodesics can be parametrized as follows. If $\lambda \in C_1$, then
$$
\begin{equation}
\begin{aligned} \, x_t&=2 k (\operatorname{cn} \varphi_t-\operatorname{cn} \varphi), \nonumber \\ y_t&=2 \bigl(\operatorname{E}(\varphi_t)-\operatorname{E}(\varphi)\bigr)-t, \nonumber \\ z_t&=2k\biggl(\operatorname{sn}\varphi_t\operatorname{dn}\varphi_t- \operatorname{sn}\varphi\operatorname{dn}\varphi- \frac{y_t}{2}(\operatorname{cn}\varphi_t+\operatorname{cn}\varphi)\biggr), \nonumber \\ v_t&=\frac{y_t^3}{6}+2 k^2 \operatorname{cn}^2 \varphi\cdot y_t- 4k^2\operatorname{cn}\varphi\cdot(\operatorname{sn}\varphi_t \operatorname{dn}\varphi_t-\operatorname{sn}\varphi\operatorname{dn}\varphi) \nonumber \\ &\qquad+2k^2\biggl(\frac{2}{3}\operatorname{cn}\varphi_t\operatorname{dn} \varphi_t \operatorname{sn}\varphi_t- \frac{2}{3} \operatorname{cn} \varphi \operatorname{dn} \varphi \operatorname{sn} \varphi+\frac{1-k^2}{3 k^2}\,t \nonumber \\ &\qquad+\frac{2 k^2 -1}{3 k^2}\bigl(\operatorname{E}(\varphi_t)- \operatorname{E}(\varphi)\bigr)\biggr). \end{aligned}
\end{equation}
\tag{2.108}
$$
If $\lambda \in C_2$, then
$$
\begin{equation}
\begin{gathered} \, \begin{aligned} \, x_t&=\frac{2\operatorname{sign}c}{k}(\operatorname{dn}\psi_t- \operatorname{dn}\psi), \nonumber \\ y_t&=\frac{k^2-2}{k^2}\,t+\frac{2}{k}\bigl(\operatorname{E}(\psi_t)- \operatorname{E}(\psi)\bigr), \nonumber \\ z_t&=-\frac{x_t y_t}{2}-\frac{2\operatorname{sign}c\, \operatorname{dn}\psi}{k}\,y_t+2\operatorname{sign}c\, (\operatorname{cn}\psi_t\operatorname{sn}\psi_t- \operatorname{cn} \psi \operatorname{sn} \psi), \nonumber \\ v_t&=\frac{4}{k}\biggl(\!\frac{1}{3}\operatorname{cn}\psi_t \operatorname{dn} \psi_t \operatorname{sn} \psi_t -\frac{1}{3} \operatorname{cn} \psi \operatorname{dn} \psi \operatorname{sn} \psi-\frac{1-k^2}{3 k^3}\,t- \frac{k^2-2}{6 k^2}\bigl(\operatorname{E}(\psi_t)- \operatorname{E}(\psi)\bigr)\!\biggr) \nonumber \\ &\qquad+\frac{y_t^3}{6}+\frac{2y_t}{k^2}\operatorname{dn}^2\psi- \frac{4}{k}\operatorname{dn}\psi\bigl(\operatorname{cn}\psi_t \operatorname{sn}\psi_t-\operatorname{cn}\psi\operatorname{sn}\psi\bigr), \nonumber \end{aligned} \\ \psi=\frac{\varphi}{k}\,, \quad \psi_t=\psi+\frac{t}{k}\,. \end{gathered}
\end{equation}
\tag{2.109}
$$
If $\lambda \in C_3$, then
$$
\begin{equation}
\begin{aligned} \, x_t&=2 \operatorname{sign}c\biggl(\frac{1}{\cosh\varphi_t}- \frac{1}{\cosh\varphi}\biggr), \nonumber \\ y_t&=2(\tanh \varphi_t-\tanh \varphi)-t , \nonumber \\ z_t&=-\frac{x_t y_t}{2}- \frac{2\operatorname{sign} c}{\cosh \varphi}\, y_t+ 2\operatorname{sign}c\biggl(\frac{\tanh\varphi_t}{\cosh\varphi_t}- \frac{\tanh\varphi}{\cosh\varphi}\biggr), \nonumber \\ v_t&=\frac{2}{3}\biggl(\tanh\varphi_t-\tanh\varphi+ 2\,\frac{\tanh\varphi_t}{\cosh^2\varphi_t}- 2\,\frac{\tanh\varphi}{\cosh^2\varphi}\biggr) \nonumber \\ &\qquad+\frac{y_t^3}{6}+\frac{2y_t}{\cosh^2\varphi}- \frac{4}{\cosh\varphi}\biggl(\frac{\tanh\varphi_t}{\cosh\varphi_t}- \frac{\tanh\varphi}{\cosh\varphi}\biggr). \end{aligned}
\end{equation}
\tag{2.110}
$$
For arbitrary $\lambda=(\varphi,k,\alpha) \in \bigcup_{i=1}^3 C_i$ a parametrization of geodesics can be obtained from the case $\alpha=1$ using dilations and reflections: In the remaining cases, when $\lambda \in \bigcup_{i=4}^7C_i$, geodesics are parametrized by elementary functions. If $\lambda \in C_4$, then
$$
\begin{equation*}
x_t=0, \qquad y_t=t \operatorname{sign}\alpha, \qquad z_t=0, \qquad v_t=\frac{t^3}{6} \operatorname{sign} \alpha.
\end{equation*}
\notag
$$
If $\lambda \in C_5$, then
$$
\begin{equation*}
x_t=0, \qquad y_t=- t \operatorname{sign} \alpha, \qquad z_t=0, \qquad v_t=-\frac{t^3}{6} \operatorname{sign} \alpha.
\end{equation*}
\notag
$$
If $\lambda \in C_6$, then
$$
\begin{equation*}
\begin{gathered} \, x_t=\frac{\cos (c t+\theta)-\cos \theta}{c}\,,\qquad y_t=\frac{\sin(c t+\theta)-\sin \theta}{c}\,, \qquad z_t=\frac{ct-\sin(ct)}{2c^2}\,, \\ v_t=\frac{3\cos\theta-2ct\sin\theta- 4\cos(ct+\theta)+\cos(2ct+\theta)}{4c^3}\,. \end{gathered}
\end{equation*}
\notag
$$
If $\lambda \in C_7$, then
$$
\begin{equation*}
x_t=-t\sin\theta, \qquad y_t=t\cos\theta, \qquad z_t=0, \qquad v_t=\frac{t^3}{6}\cos\theta.
\end{equation*}
\notag
$$
The projections of geodesics onto the $(x,y)$-plane are Euler elasticae (see § 2.6), namely, inflectional ones for $\lambda \in C_1$, non-inflectional ones for $\lambda \in C_2$, critical ones for $\lambda \in C_3$, straight lines for $\lambda \in C_4 \cup C_5 \cup C_7$, and circles for $\lambda \in C_6$. The family of all geodesics is parametrized by means of the exponential map
$$
\begin{equation*}
\operatorname{Exp} \colon N=C \times \mathbb{R}_+ \to M,\quad \operatorname{Exp}(\lambda,t)=g_t=(x_t,y_t,z_t,v_t).
\end{equation*}
\notag
$$
2.9.4. Symmetries of the exponential map and the Maxwell timeThe dilations (2.106), (2.107) form a one-parameter group of symmetries of the exponential map. It also has the discrete group of symmetries formed by reflections,
$$
\begin{equation*}
\operatorname{Sym}=\{\operatorname{Id},\varepsilon^1,\dots,\varepsilon^7\} \cong \mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2.
\end{equation*}
\notag
$$
Let $\vec H_v=c\,\dfrac{\partial}{\partial\theta}- \alpha\sin\theta\,\dfrac{\partial}{\partial c}\in\operatorname{Vec}(C)$ be the vertical component of the normal Hamiltonian field $\vec H$. The maps $\varepsilon^i\colon C \to C$ below preserve the direction field of the vector field $\vec H_v$:
$$
\begin{equation*}
\begin{aligned} \, \varepsilon^1\colon (\theta,c,\alpha) &\mapsto (\theta,-c,\alpha), \\ \varepsilon^2\colon (\theta,c,\alpha) &\mapsto (-\theta,c,\alpha), \\ \varepsilon^3\colon (\theta,c,\alpha) &\mapsto (-\theta,-c,\alpha), \\ \varepsilon^4\colon (\theta,c,\alpha) &\mapsto (\theta+\pi,c,-\alpha), \\ \varepsilon^5\colon (\theta,c,\alpha) &\mapsto (\theta+\pi,-c,-\alpha), \\ \varepsilon^6\colon (\theta,c,\alpha) &\mapsto (-\theta+\pi,c,-\alpha), \\ \varepsilon^7\colon (\theta,c,\alpha) &\mapsto (-\theta+\pi,-c,-\alpha). \end{aligned}
\end{equation*}
\notag
$$
Namely, $\varepsilon^i_*\vec H_v=\vec H_v$ for $i=3,4,7$ and $\varepsilon^i_*\vec H_v=-\vec H_v$ for $i=1,2,5,6$. The actions of the reflections $\varepsilon^i\colon C \to C$ extend to symmetries of the exponential map as follows. The action $\varepsilon^i\colon N \to N$, $N=C \times \mathbb{R}_{+}$, is defined by
$$
\begin{equation*}
{\varepsilon}^i(\lambda,t)=\begin{cases} \bigl({\varepsilon}^i(\lambda),t\bigr) & \text{if}\ {\varepsilon}^i_* \vec{H}_v=\vec{H}_v, \\ \bigl({\varepsilon}^i \circ e^{t \vec{H}_v}(\lambda),t\bigr) & \text{if } \ {\varepsilon}^i_* \vec{H}_v=-\vec{H}_v. \end{cases}
\end{equation*}
\notag
$$
The action $\varepsilon^i\colon G \to G$ is defined by
$$
\begin{equation*}
\varepsilon^i(q)=\varepsilon^i(x,y,z,v)=g^i=(x^i,y^i,z^i,v^i),
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{aligned} \, (x^1, y^1, z^1, v^1)&=(x, y, -z, v-x z), \\ (x^2, y^2, z^2, v^2)&=(-x, y, z, v-x z), \\ (x^3, y^3, z^3, v^3)&=(-x, y, -z, v), \\ (x^4, y^4, z^4, v^4)&=(-x, -y, z, -v), \\ (x^5, y^5, z^5, v^5)&=(-x, -y, -z, -v+x z), \\ (x^6, y^6, z^6, v^6)&=(x, -y, z, -v+x z), \\ (x^7, y^7, z^7, v^7)&=(x, -y, -z, -v). \end{aligned}
\end{equation*}
\notag
$$
Proposition 2.11. The group $\operatorname{Sym}$ is a subgroup of the symmetry group of the exponential map. Theorem 2.41. On almost all geodesics the first Maxwell time corresponding to the symmetry group $\operatorname{Sym}$ can be expressed as follows: Remark 2.3. On geodesics with first Maxwell time distinct from $t_{\rm Max}^1$, it is lager than this quantity, while $t_{\rm Max}^1$ is the first conjugate time. Theorem 2.42. The function $t_{\rm Max}^1\colon C \to (0,+\infty]$ has the following invariance properties: (1) $t_{\rm Max}^1(\lambda)$ depends only on $E$ and $|\alpha|$; (2) $t_{\rm Max}^1(\lambda)$ in a first integral of the field $\vec H_v$; (3) $t_{\rm Max}^1(\lambda)$ is reflection invariant: if $(\lambda,t) \in C\times \mathbb{R}_+$, $(\lambda^i, t)=\varepsilon^i(\lambda, t)$, then (4) $t_{\rm Max}^1(\lambda)$ is dilation homogeneous: if $\lambda \in C$ and $\lambda_s=\delta_s(\lambda) \in C$, then 2.9.5. A lower bound for the conjugate time2.9.6. The diffeomorphism structure of the exponential mapConsider the following subset of the state space, which does not contain fixed points of the symmetries $\varepsilon^1$ and $\varepsilon^2$:
$$
\begin{equation*}
\widetilde G=\{g \in G \mid \varepsilon^1(g) \ne g \ne \varepsilon^2(g)\}= \{g \in G \mid xz \ne 0\},
\end{equation*}
\notag
$$
and consider its connected components
$$
\begin{equation*}
\begin{gathered} \, G_1=\{g \in G \mid x < 0, z > 0\}, \\ G_2=\{g \in G \mid x < 0, z < 0\}, \\ G_3=\{g \in G \mid x > 0, z < 0\}, \end{gathered}
\end{equation*}
\notag
$$
and Also consider the open dense subset of the space of potentially optimal geodesics
$$
\begin{equation*}
\widetilde{N}=\{(\lambda, t) \in N \mid t < t_{\rm Max}^1(\lambda), \ c_{t/2} \sin \theta_{t/2} \ne 0\}
\end{equation*}
\notag
$$
and its connected components
$$
\begin{equation*}
\begin{gathered} \, D_1=\bigl\{(\lambda, t) \in N \mid t \in \bigl(0,t_{\rm Max}^1(\lambda)\bigr), \ \sin\theta_{t/2} > 0, \ c_{t/2} > 0\bigr\}, \\ D_2=\bigl\{(\lambda, t) \in N \mid t \in \bigl(0,t_{\rm Max}^1(\lambda)\bigr), \ \sin\theta_{t/2} > 0, \ c_{t/2} < 0\bigr\}, \\ D_3=\bigl\{(\lambda, t) \in N \mid t \in \bigl(0, t_{\rm Max}^1(\lambda)\bigr), \ \sin\theta_{t/2} < 0, \ c_{t/2} < 0\bigr\}, \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
D_4=\bigl\{(\lambda, t) \in N \mid t \in \bigl(0, t_{\rm Max}^1(\lambda)\bigr), \ \sin\theta_{t/2} < 0, \ c_{t/2} > 0\bigr\}.
\end{equation*}
\notag
$$
2.9.7. The cut time2.9.8. The cut locus and its stratificationTheorem 2.46. The cut locus $\operatorname{Cut}$ lies in the union of coordinate subspaces $\{x=0\}$ and $\{z=0\}$. It is invariant under dilations and discrete symmetries: Theorem 2.47. The cut locus has the stratification The three-dimensional strata $\mathcal{I}_{x\pm}$ and $\mathcal{I}_{z\pm}$ (or $\mathcal{N}_{x\pm}$) consist of points such that the projections of length minimizers onto the $(x,y)$-plane are elasticae, which are inflectional, that is, with inflection points (or non-inflectional, that is, without inflection points, respectively); see § 2.6. For the one-dimensional strata $\mathcal{E}_{\pm}$ the corresponding elastica are closed (have the shape of a ‘figure-of-eight’). In Figs 35 and 36 we show the stratifications of the cut locus and its intersection with a coordinate subspace. In Fig. 35 the reader can see the adjacency topology of the quotients of strata of the cut locus by the dilations $Y$. In Fig. 36 we show the intersection $\operatorname{Cut} \mathrel{\cap} \{x=z=0\}$. In Fig. 37 we show the quotient of the set $\operatorname{Cut}\cap \{z=0\}$ by the dilations $Y$; the quotient $\{z=0\}/ e^{\mathbb{R} Y}$ is shown there as the topological sphere $\{g \in G \mid x^6+y^6+w^2=1\}$. In a similar way, in Fig. 38 the quotient $(\operatorname{Cut}\mathrel{\cap} \{x=0\})/ e^{\mathbb{R}Y}$ is shown on the topological sphere $\{g \in G \mid y^6+|z|^3+w^2=1\}$. It is obvious that a unique sub-Riemannian length minimizer comes into each point $g_1 \in G \setminus \operatorname{Cut}$. Below we describe a similar property of points $g_1 \in \operatorname{Cut}$. Theorem 2.48. (1) Precisely two length minimizers come into each point in the three- dimensional strata of the cut locus (these strata consist of Maxwell points that are not conjugate points). (2) A unique length minimizer comes into each point in the two-dimensional strata (these strata consist of conjugate points that are not Maxwell points). (3) A one-parameter family of length minimizers come into each point in the one- dimensional strata (these strata consist of Maxwell points that are incidentally conjugate points). The cut locus is not closed because it contains points lying arbitrarily close to the reference point $q_0$, but not elasticae itself (this is a general fact of sub-Riemannian geometry). The closure of the cut locus in the sub-Riemannian problem on the Engel group has the following simple description. Theorem 2.49. The equality $\operatorname{cl}(\operatorname{Cut})=\operatorname{Cut} \mathrel{\sqcup} \mathcal{A}_+ \sqcup \mathcal{A}_- \sqcup \{g_0\}$ holds. The abnormal trajectories $\mathcal{A}_{\pm}$ adjoin strata of the cut locus as shown in Fig. 35. 2.9.9. SpheresSub-Riemannian spheres are taken to one another by left shifts: and by dilations:
$$
\begin{equation*}
\delta_s(S_R(\operatorname{Id}))=S_{R'}(\operatorname{Id}), \qquad R'=e^s R,
\end{equation*}
\notag
$$
so it is sufficient to examine the unit sphere $S=S_1(\operatorname{Id})$. The unit sphere is invariant under reflections: Consider the cross-section of the unit sphere by the two-dimensional fixed-point manifold of the basic symmetries $\varepsilon^1$ and $\varepsilon^2$:
$$
\begin{equation*}
\widetilde{S}=\{g \in S \mid \varepsilon^1(g)= \varepsilon^2(g)=g\}=S \cap \{x=z=0\}
\end{equation*}
\notag
$$
(see Fig. 39). The cross-section $\widetilde{S}$ is centrally symmetric because of the reflection $\varepsilon^4$:
$$
\begin{equation*}
\begin{gathered} \, \varepsilon^4(\gamma_i)=\gamma_{i+2}, \qquad i=1, 2, \\ \varepsilon^4(A_+)=A_-, \qquad \varepsilon^4(C_+)=C_-. \end{gathered}
\end{equation*}
\notag
$$
Different points in $\widetilde{S}$ can be characterized as follows:
Theorem 2.51. The cross-section $\widetilde{S}$ has the following regularity at different points: (1) the curves $\gamma_i$ are analytic and regular; (2) $A_{\pm}$ and $C_{\pm}$ are singular points; $\widetilde{S}$ is not smooth but Lipschitz at these points; (3) $\overline{\gamma}_2=\gamma_2 \cup \{C_+, A_+\}$ is smooth of class $C^{\infty}$; (4) $\gamma_1 \cup \{C_+\}$ is smooth of class $C^{\infty}$; (5) $\gamma_1 \cup \{A_-\}$ is smooth of class $C^{1}$. Theorem 2.52. (1) The set $\widetilde{S} \setminus \{A_+, A_-\}$ is semianalytic and therefore subanalytic. (2) In a neighbourhood of $A_-$ the curve $\gamma_1$ is the graph of a non-analytic function
$$
\begin{equation*}
w=\frac 16 Y^3-4 Y^3 \exp\biggl(-\frac{2}{Y}\biggr)(1+ o(1)), \qquad Y=\frac{y+1}{2} \to 0.
\end{equation*}
\notag
$$
(3) The set $\widetilde{S}$ is not semianalytic, so it is not subanalytic. (4) The sphere $S$ is not subanalytic. Remark 2.4. That the Engel sphere $S$ is not subanalytic also follows from its projection onto the (non-subanalytic) Martinet sphere (see § 2.3). Theorem 2.53. In a neighbourhood of $A_-$ the curve $\gamma_1$ is the graph of a function in the $\exp$-$\log$-category, where $F(\xi,\eta)$ is an analytic function in a neighbourhood of the point $(\xi,\eta )=(0,0)$. Hence $\widetilde{S}$ is a set in the $\exp$-$\log$-category. 2.9.10. Explicit expressions for the sub-Riemannian distanceFor some points in the Engel group we know their distances to the identity element:
2.9.11. Metric linesTheorem 2.55. The following geodesics and only they are metric straight lines with natural parametrization on the Engel group: (1) one-parameter subgroups tangent to the distribution:
$$
\begin{equation}
\begin{aligned} \, &e^{(u_1X_1+u_2X_2) t}=\operatorname{Exp}(\lambda, t), \qquad t \in \mathbb{R}, \\ &u_1=-\sin \theta, \quad u_2=\cos \theta, \quad \lambda=(\theta, c=0, \alpha) \in C_4\cup C_5, \nonumber \end{aligned}
\end{equation}
\tag{2.115}
$$
(2) critical geodesics: Remark 2.5. The geodesics (2.115) project onto the $(x,y)$-plane as Euclidean straight lines. Among them only the curves 2.9.12. Bibliographic commentsSubsections 2.9.1, 2.9.3, and 2.9.4 are based on [19], § 2.9.2 is based on [124], § 2.9.5 on [20], §§ 2.9.6 and 2.9.11 on [21], § 2.9.8 on [22], and § 2.9.9 on [145]. A parametization of the Riemannian geodesics on the Engel group was originally obtained in [158]. The sub-Riemannian problem on the Engel group was also considered in [23] and [142]. 2.10. The sub-Riemannian problem on the Cartan group2.10.1. The statement of the problemThe geometric setting. Consider the following generalization (sophistication) of problems on the Heisenberg group [143], [157] and Engel group (§ 2.9): the generalized Dido problem. Let $a_0,a_1 \in \mathbb{R}^2$ be points in the plane which are joined by a curve $\gamma_0 \subset \mathbb{R}^2$. Also fix $S \in \mathbb{R}$ and $c\in \mathbb{R}^2$. The problem consists in connecting $a_0$ with $a_1$ by a shortest curve $\gamma \subset \mathbb{R}^2$ such that the two curves $\gamma_0$ and $\gamma$ bound a plane domain of algebraic area $S$ with centre of mass $c$. The optimal control problem. We can also state this geometric problem as an optimal control problem:
$$
\begin{equation}
\dot{g}=u_1 X_1(g)+u_2 X_2(g), \qquad g=(x, y, z, v, w) \in \mathbb{R}^5,
\end{equation}
\tag{2.117}
$$
$$
\begin{equation}
X_1=\frac{\partial}{\partial x}-\frac{y}{2}\,\frac{\partial}{\partial z}- \frac{x^2 +y^2}{2}\,\frac{\partial}{\partial w}\,, \quad X_2=\frac{\partial}{\partial y}+\frac{x}{2}\,\frac{\partial}{\partial z}+ \frac{x^2 +y^2}{2}\,\frac{\partial}{\partial v}\,.
\end{equation}
\tag{2.120}
$$
This is a sub-Riemannian problem for the sub-Riemannian structure on $\mathbb{R}^5$ defined by the vector fields $X_1$ and $X_2$ as an orthonormal frame. The Cartan algebra and Cartan group. The Cartan algebra if the free five- dimensional nilpotent Lie algebra $\mathfrak{g}$ with two generators which has depth 3. There exists a basis $\mathfrak{g}=\operatorname{span}(X_1,\dots,X_5)$ such that the nontrivial Lie brackets in this basis are
$$
\begin{equation*}
[X_1,X_2]=X_3, \quad [X_1,X_3]=X_4, \quad\text{and}\quad [X_2,X_3]=X_5
\end{equation*}
\notag
$$
(see Fig. 40). The Cartan algebra has the grading
$$
\begin{equation*}
\mathfrak{g}=\mathfrak{g}^{(1)} \oplus \mathfrak{g}^{(2)} \oplus \mathfrak{g}^{(3)},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{gathered} \, \mathfrak{g}^{(1)}=\operatorname{span}(X_1,X_2),\quad \mathfrak{g}^{(2)}=\mathbb{R} X_3,\quad \mathfrak{g}^{(3)}=\operatorname{span}(X_4,X_5),\qquad [\mathfrak{g}^{(1)}, \mathfrak{g}^{(i)}]=\mathfrak{g}^{(i+1)}, \\ \text{and}\quad \mathfrak{g}^{(4)}=\mathfrak{g}^{(5)}=\{0\}, \end{gathered}
\end{equation*}
\notag
$$
so it is a Carnot algebra. The corresponding connected and simply connected Lie group $G$ is called the Cartan group. On the space $\mathbb{R}^5_{x,y,z,v,w}$ we can introduce multiplication by
$$
\begin{equation*}
\begin{pmatrix} x_1 \\ y_1 \\ z_1 \\ v_1 \\ w_1\end{pmatrix}\cdot \begin{pmatrix} x_2 \\ y_2 \\ z_2 \\ v_2 \\ w_2\end{pmatrix}= \begin{pmatrix} x_1+x_2 \\ y_1+y_2 \\ z_1+z_2+(x_1 y_2-y_1 x_2)/2 \\ v_1+v_2+(x_1^2+y_1^2+x_1 x_2+y_1 y_2)y_2/2+x_1 z_2 \\ w_1+w_2-(x_1^2+y_1^2+x_1x_2+y_1 y_2)x_2/2+y_1z_2 \end{pmatrix},
\end{equation*}
\notag
$$
which makes of it the Cartan group, $G \cong \mathbb{R}^5_{x,y,z,v,w}$, so that the fields (2.120) are left invariant on this group. Thus, problem (2.117)–(2.119) is a left-invariant sub-Riemannian problem on the Cartan group. Hence we can assume that $g_0=\operatorname{Id}=(0,\dots,0)$. Apart from the model (2.120), there are also other models of the sub-Riemannian problem on the Cartan group [10], [56], [124]. The left-invariant sub-Riemannian problem with growth vector $(2,3,5)$ on the Cartan group is unique up to an automorphism of this group [124]. Special features of the problem. The sub-Riemannian problem on the Cartan group is the simplest left-invariant problem with the following properties:
This is the unique free nilpotent sub-Riemannian problem of depth 3 with Liouville-integrable normal Hamiltonian field of the Pontryagin maximum principle (Liouville non-integrable problems are free nilpotent problems of depth 3 and rank greater than 2 [50] and ones of depth greater than 3 and rank at least 2 [104]). The distribution $\Delta=\operatorname{span}(X_1,X_2)$ has a 14-dimensional algebra of infinitesimal symmetries, which is the special Lie algebra $\mathfrak{g}_2$: this fact goes back to È, Cartan’s famous ‘five-dimensional’ paper [62] (also see § 2.10.2 below). Finally, the sub-Riemannian problem on the Cartan group provides a nilpotent approximation to any problem with growth vector $(2,3,5)$, for instance,
2.10.2. Symmetries of the distribution and sub-Riemannian structureTheorem 2.56. The Lie algebra of infinitesimal symmetries of the distribution $\Delta$ on the Cartan group is the $14$-dimensions algebra $\mathfrak{g}_2$ that is the non-compact real form of the special complex Lie algebra $\mathfrak{g}_2^{\mathbb{C}}$. Theorem 2.57. The Lie algebra of infinitesimal symmetries of the nilpotent sub- Riemannian structure on the Cartan group is the $6$-dimensional Lie algebra in which there exists a basis $X_0,Y_1,\dots,Y_5$ with non-trivial brackets A representation of the Lie algebra of symmetries of the distribution and sub- Riemannian structure by vector fields in $\mathbb{R}^5$ was presented in [124]. 2.10.3. GeodesicsThe existence of optimal controls in problem (2.117)–(2.119) is a consequence of the Rashevskii–Chow and Filippov theorems. The Pontryagin maximum principle. Going over from the minimization of length (2.119) to the minimization of the energy $J=\dfrac{1}{2}\displaystyle\int_0^{t_1} (u_1^2+u_2^2)\,dt$ and using the Hamiltonians $h_i(\lambda)=\langle\lambda, X_i\rangle$, $i=1,\dots,5$, which are linear on fibres of $T^*G$ we obtain the assumptions of the Pontryagin maximum principle
$$
\begin{equation*}
\begin{gathered} \, \dot h_1=-u_2 h_3,\quad \dot h_2=u_1 h_3,\quad \dot h_3=u_1 h_4+u_2 h_5,\quad \dot h_4=0,\quad \dot h_5=0, \\ \dot{g}=u_1 X_1+u_2 X_2, \\ u_1 h_1+u_2 h_2+\frac{\nu}{2}(u_1^2+u_2^2) \to \max_{(u_1,u_2) \in \mathbb{R}^2}, \\ \nu \leqslant 0,\qquad (h_1,\dots,h_5,\nu) \ne 0. \end{gathered}
\end{equation*}
\notag
$$
Abnormal extremals. Abnormal extremals of constant velocity can be parametrized as follows:
$$
\begin{equation}
h_1= h_2=h_3=0, \quad (h_4, h_5) \equiv \operatorname{const} \ne 0, \nonumber
\end{equation}
\notag
$$
Abnormal trajectories (2.121)–(2.125) are one-parameter subgroups $g_t=e^{t(u_1X_1+u_2X_2)}$ tangent to the distribution $\Delta$. They project onto the $(x,y)$-planes as straight lines, so they are length minimizers. The abnormal set is a smooth two-dimensional manifold diffeomorphic to $\mathbb{R}^2$:
$$
\begin{equation*}
\operatorname{Abn}=\biggl\{g \in G \Bigm| z=v-\frac{1}{6}(x^2+y^2)x= w+\frac{1}{6}(x^2+y^2)y=0\biggr\}.
\end{equation*}
\notag
$$
Normal extremals. Normal extremals satisfy the Hamiltonian system
$$
\begin{equation}
\dot{\lambda}=\vec{H}(\lambda), \quad \lambda \in T^*G,
\end{equation}
\tag{2.126}
$$
with Hamiltonian $H=(h_1^2+h_2^2)/2$. On the level surface $\{H=1/2\}$ we introduce the coordinates $(\theta,c,\alpha,\beta) \in S^1\times\mathbb{R}\times\mathbb{R}_+\times S^1$ such that
$$
\begin{equation}
h_1=\cos \theta, \quad h_2=\sin \theta, \quad h_3=c, \quad h_4=\alpha \sin \beta, \quad h_5=-\alpha \cos \beta.
\end{equation}
\tag{2.127}
$$
Then the Hamiltonian system (2.126) becomes
$$
\begin{equation}
\dot{\theta}=c, \quad \dot c=-\alpha \sin(\theta-\beta), \quad \dot{\alpha}=\dot{\beta}=0,
\end{equation}
\tag{2.128}
$$
The vertical subsystem (2.128) is the pendulum equations. The projections of normal geodesics onto the $(x,y)$-plane are Euler elasticae (see § 2.6). Abnormal length minimizers (2.121)–(2.125) satisfy the normal Hamiltonian system (2.128), (2.129) for $\theta=\beta$ and $c=0$, so they are non-strictly abnormal. The symplectic foliation and Casimir functions. There exist three independent Casimir functions on the Lie coalgebra $\mathfrak{g}^*$:
$$
\begin{equation*}
h_4, \quad h_5, \quad E=\frac{h_3^2}{2}+h_1 h_5-h_2 h_4.
\end{equation*}
\notag
$$
The symplectic foliation on $\mathfrak{g}^*$ consists of
Parametrization of normal geodesics. The family of normal extremals on the level surface $\{H=1/2\}$ is parametrized by their initial points on the cylinder This cylinder is stratified depending on different types of trajectories of the pendulum (2.128): where
$$
\begin{equation*}
\begin{gathered} \, C_1=\{\lambda \in C \mid \alpha > 0, \ E \in (-\alpha,\alpha)\}, \\ C_2=\{\lambda \in C \mid \alpha > 0, \ E \in (\alpha,+\infty)\}, \\ C_3=\{\lambda \in C \mid \alpha > 0, \ E=\alpha, \ \theta-\beta \ne \pi\}, \\ C_4=\{\lambda \in C \mid \alpha > 0, \ E=-\alpha\}, \\ C_5=\{\lambda \in C \mid \alpha > 0, \ E=\alpha, \ \theta-\beta=\pi\}, \\ C_6=\{\lambda \in C \mid \alpha=0, \ c \ne 0\}, \end{gathered}
\end{equation*}
\notag
$$
and To parametrize normal geodesics, on the strata $C_1$, $C_2$, and $C_3$ we introduce elliptic coordinates $(\varphi,k,\alpha,\beta)$ in which the pendulum equation (2.128) straightens out: if $\lambda \in C_1$, then
$$
\begin{equation*}
\begin{gathered} \, k=\sqrt{\frac{E+\alpha}{2\alpha}}= \sqrt{\sin^2\frac{\theta-\beta}{2}+\frac{c^2}{4\alpha}} \in (0,1), \qquad \varphi \in [0,4K], \\ \begin{cases} \sin\dfrac{\theta-\beta}{2}=k\operatorname{sn}(\sqrt{\alpha}\,\varphi), \\ \dfrac{c}{2}=k\sqrt{\alpha}\,\operatorname{cn}(\sqrt{\alpha}\,\varphi); \end{cases} \end{gathered}
\end{equation*}
\notag
$$
if $\lambda \in C_2$, then
$$
\begin{equation*}
\begin{gathered} \, k=\sqrt{\frac{2\alpha}{E+\alpha}}=\biggl(\sqrt{\sin^2\frac{\theta-\beta}{2}+ \frac{c^2}{4\alpha}}\biggr)^{-1}\in (0, 1),\qquad \varphi \in [0,2kK], \\ \begin{cases} \sin\dfrac{\theta-\beta}{2}=\pm \operatorname{sn} \dfrac{\sqrt{\alpha}\,\varphi}{k}\,, \\ \dfrac{c}{2}=\pm \dfrac{\sqrt{\alpha}}{k} \operatorname{dn} \dfrac{\sqrt{\alpha}\,\varphi}{k}\,, \end{cases} \qquad \pm=\operatorname{sign} c, \\ \psi=\frac{\varphi}{k}\,; \end{gathered}
\end{equation*}
\notag
$$
if $\lambda \in C_3$, then
$$
\begin{equation*}
\begin{gathered} \, k=1, \qquad \varphi \in (-\infty,+\infty), \\ \begin{cases} \sin\dfrac{\theta-\beta}{2}=\pm\tanh(\sqrt{\alpha}\,\varphi), \\ \dfrac{c}{2}=\pm\dfrac{\sqrt{\alpha}}{\cosh(\sqrt{\alpha}\,\varphi)}\,, \end{cases} \qquad \pm=\operatorname{sign} c. \end{gathered}
\end{equation*}
\notag
$$
Then
$$
\begin{equation*}
\dot{\varphi}=1, \quad \dot{k}=\dot{\alpha}=\dot{\beta}=0.
\end{equation*}
\notag
$$
The problem is invariant under left shifts on the Cartan group, the dilations
$$
\begin{equation*}
\begin{gathered} \, e^{sY}\!\colon (t,x,y,z,v,w)\mapsto (e^st,e^sx,e^sy,e^{2s}z,e^{3s}v,e^{3s}w), \\ (\theta,c,\alpha,\beta)\mapsto (\theta,e^{-s}c,e^{-2s}\alpha,\beta), \\ (\varphi,k,\alpha,\beta)\mapsto (e^s\varphi,k,e^{-2s}\alpha,\beta), \\ Y=x\,\frac{\partial}{\partial x}+y\,\frac{\partial}{\partial y}+ 2z\,\frac{\partial}{\partial z}+3v\,\frac{\partial}{\partial v}+ 3w\,\frac{\partial}{\partial w}\,, \end{gathered}
\end{equation*}
\notag
$$
and the rotations
$$
\begin{equation}
\begin{aligned} \, e^{rX_0}\colon (x,y,z,v,w)&\mapsto (x\cos r-y\sin r,x\sin r+y\cos r,z, \nonumber \\ &\qquad v\cos r-w\sin r,v\sin r+w\cos r). \end{aligned}
\end{equation}
\tag{2.130}
$$
Using dilations and rotations we can take any covector $\lambda=(\varphi,k,\alpha,\beta) \in\bigcup_{i=1}^3C_i$ to the fundamental set $\{\alpha=1, \ \beta=0\}$. For $\alpha=1$, $\beta=0$, and $\lambda \in\bigcup_{i=1}^3C_i$ geodesics $g_t=(x_t,y_t,z_t,v_t,w_t)$ are parametrized as follows. If $\lambda \in C_1$, then
$$
\begin{equation*}
\begin{gathered} \, x_t=2(\operatorname{E}(\varphi_t)-\operatorname{E}(\varphi))- (\varphi_t-\varphi), \\ y_t=2k(\operatorname{cn} \varphi-\operatorname{cn} \varphi_t), \\ z_t=2k(\operatorname{sn} \varphi_t \operatorname{dn} \varphi_t- \operatorname{sn}\varphi\operatorname{dn}\varphi)- k(\operatorname{cn}\varphi+\operatorname{cn}\varphi_t)x_t, \\ v_t=2k \operatorname{sn} \varphi_t \operatorname{dn} \varphi_t\cdot x_t- k \operatorname{cn} \varphi_t\cdot x_t^2-(1-2k^2+ 2k^2 \operatorname{cn} \varphi \operatorname{cn}\varphi_t) y_t, \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, w_t&=-\frac{1}{6}\bigl[x_t^3+2(2k^2-1+6k^2\operatorname{cn}^2\varphi)x_t+ 2(\varphi_t-\varphi) \\ &\qquad+8k^2(\operatorname{sn} \varphi_t \operatorname{cn} \varphi_t \operatorname{dn} \varphi_t-\operatorname{sn} \varphi\operatorname{cn} \varphi \operatorname{dn} \varphi) \\ &\qquad-24 k^2 \operatorname{cn} \varphi (\operatorname{sn} \varphi_t \operatorname{dn} \varphi_t- \operatorname{sn} \varphi \operatorname{dn} \varphi)\bigr], \end{aligned}
\end{equation*}
\notag
$$
where $\varphi_t=\varphi+t$. If $\lambda \in C_2$, then
$$
\begin{equation*}
\begin{gathered} \, x_t=\frac{2}{k}\biggl(\operatorname{E}(\psi_t)- \operatorname{E}(\psi)-\frac{2-k^2}{2}(\psi_t-\psi)\biggr), \\ y_t=\pm \frac{2}{k}(\operatorname{dn} \psi-\operatorname{dn} \psi_t), \\ z_t=\pm \biggl(2(\operatorname{sn} \psi_t \operatorname{cn}\psi_t- \operatorname{sn} \psi \operatorname{cn} \psi)- \frac{1}{k}(\operatorname{dn}\psi+\operatorname{dn}\psi_t)x_t\biggr), \\ v_t=\pm \biggl(2\operatorname{sn} \psi_t \operatorname{cn} \psi_t\cdot x_t- \frac{1}{k} \operatorname{dn} \psi_t\cdot x_t^2 \biggr)+ \frac{1}{k^2}(2-k^2-2\operatorname{dn} \psi \operatorname{dn} \psi_t) y_t, \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, w_t&=-\frac{1}{6}\biggl(x_t^3+\frac{2}{k^2}(2-k^2+ 6 \operatorname{dn}^2 \psi) x_t+2 k(\psi_t-\psi) \\ &\qquad+\frac{8}{k}(\operatorname{sn}\psi_t\operatorname{cn} \psi_t \operatorname{dn}\psi_t-\operatorname{sn} \psi \operatorname{cn} \psi \operatorname{dn}\psi) \\ &\qquad-\frac{24}{k}\operatorname{dn}\psi\,(\operatorname{sn}\psi_t \operatorname{cn}\psi_t-\operatorname{sn}\psi\operatorname{cn}\psi)\biggr), \end{aligned}
\end{equation*}
\notag
$$
where $\pm=\operatorname{sign}c$ and $\psi_t=\psi+t/k$. If $\lambda \in C_3$, then
$$
\begin{equation*}
\begin{gathered} \, x_t=2(\tanh \varphi_t-\tanh\varphi)-(\varphi_t-\varphi), \\ y_t=\pm 2 \biggl(\frac{1}{\cosh\varphi}-\frac{1}{\cosh\varphi_t}\biggr), \\ z_t=\pm \biggl( 2 \biggl( \frac{\operatorname{sinh} \varphi_t} {\cosh^2 \varphi_t}-\frac{\operatorname{sinh}\varphi}{\cosh^2 \varphi} \biggr)- \biggl( \frac{1}{\cosh \varphi}+\frac{1}{\cosh \varphi_t} \biggr) x_t \biggr), \\ v_t=\pm \biggl(\frac{2}{\operatorname{sinh} \varphi_t}\,x_t- \frac{1}{\cosh \varphi_t}\,x_t^2\biggr)+ \biggl(1-\frac{2}{\cosh \varphi \cosh \varphi_t}\biggr) y_t, \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, w_t&=-\frac{1}{6}\biggl(x_t^3+ 6\,\frac{2+\cosh^2\varphi}{\cosh^2\varphi}\,x_t+6(\varphi_t-\varphi) \\ &\qquad-\frac{24}{\cosh\varphi}\biggl(\frac{\operatorname{sinh} \varphi_t} {\cosh^2 \varphi_t}-\frac{\operatorname{sinh} \varphi}{\cosh^2 \varphi}\biggr)- 8(\tanh^3 \varphi_t-\tanh^3 \varphi) \biggr), \end{aligned}
\end{equation*}
\notag
$$
where $\pm=\operatorname{sign} c$ and $\varphi_t=\varphi+t$. We obtain a parametrization of geodesics for arbitrary $\lambda=(\varphi,k,\alpha,\beta) \in\bigcup_{i=1}^3C_i$ from the case where $\alpha=1$ and $\beta=0$ by use of rotations and dilations:
$$
\begin{equation*}
\begin{gathered} \, g_t(\varphi,k,\alpha,\beta)=e^{-rX_0}\circ e^{-sY}(g_{t'}(\varphi',k,\alpha'=1,\beta'=0)), \\ t'=t\sqrt{\alpha}\,, \quad \varphi'=\varphi\sqrt{\alpha}\,, \quad r=-\beta, \quad s=\frac{1}{2} \log \alpha. \end{gathered}
\end{equation*}
\notag
$$
In the remaining cases, where $\lambda=(\theta,c,\alpha,\beta) \in \bigcup_{i=4}^7C_i$, geodesics are parametrized by elementary functions: if $\lambda=(\theta,c,\alpha,\beta) \in C_4 \cup C_5 \cup C_7$ and $\beta=0$, then
$$
\begin{equation*}
(x_t,y_t,z_t,v_t,w_t)=\biggl(t,0,0,0,-\frac{t^3}{6}\biggr);
\end{equation*}
\notag
$$
while in the general case, where $\lambda \in C_4 \cup C_5 \cup C_7$,
$$
\begin{equation*}
g_t(\theta,c,\alpha,\beta)= e^{-r X_0}\bigl(g_t(\theta',c,\alpha,\beta'=0)\bigr), \quad \theta'=\theta-\beta, \quad r=-\beta;
\end{equation*}
\notag
$$
if $\lambda=(\theta=0,c,\alpha=0) \in C_6$, then
$$
\begin{equation*}
\begin{gathered} \, x_t=\frac{\sin \tau}{c}\,, \qquad y_t=\frac{1-\cos \tau}{c}\,, \qquad z_t=\frac{\tau-\sin \tau}{2c^2}\,, \\ v_t=\frac{\cos(2\tau)-4 \cos \tau+3}{4c^3}\,, \qquad w_t=\frac{\sin(2\tau)-4 \sin \tau+2\tau}{4c^3}\,, \qquad \tau=ct; \end{gathered}
\end{equation*}
\notag
$$
while in the general case where $\lambda \in C_6$,
$$
\begin{equation*}
g_t(\theta,c,\alpha=0,t)=e^{\theta X_0}(g_t(\theta' =0,c,\alpha=0,t)).
\end{equation*}
\notag
$$
The family of all geodesics is parametrized by means of the exponential map:
$$
\begin{equation*}
\operatorname{Exp}\colon(\lambda,t) \mapsto g_t=\pi \circ e^{t \vec H}(\lambda), \quad C \times \mathbb{R}_+ \to G.
\end{equation*}
\notag
$$
2.10.4. Symmetries and Maxwell strataContinuous symmetries. Dilations and rotations form a two-parameter group of continuous symmetries of the exponential map. Consider the Haimltonians
$$
\begin{equation*}
h_0(\lambda)=\langle \lambda,X_0(g) \rangle \quad\text{and}\quad h_Y(\lambda)=\langle \lambda,Y(g) \rangle,\qquad \lambda \in T^*G,
\end{equation*}
\notag
$$
which are linear on fibres of $T^*G$, and consider the corresponding Hamiltonian vector fields Then
$$
\begin{equation*}
\begin{alignedat}{2} [\vec h_0, \vec H]&=0, &\qquad \vec h_0 H&=0, \\ [\vec h_Y, \vec H]&=-2 \vec H, &\qquad \vec h_Y H&=-2 H. \end{alignedat}
\end{equation*}
\notag
$$
Let $e=\displaystyle\sum_{i=1}^5 h_i\,\dfrac{\partial}{\partial h_i}$ be a vertical Euler field on $T^*G$. As $H$ is quadratic on fibres of $T^*G$ , the Hamiltonian field $\vec H$ is linear on fibres, so that Hence the vector field $Z=\vec h_Y+e$ satisfies the equalities
$$
\begin{equation}
\nonumber [Z,\vec H]=-\vec H \quad\text{and}\quad Z H=0.
\end{equation}
\notag
$$
Moreover, Discrete symmetries. The quotient of the vertical system (2.128) by the rotations $X_0$ and dilations $Y$ is the standard pendulum equation
$$
\begin{equation*}
\dot \theta=c, \quad \dot c=-\sin \theta, \qquad (\theta,c) \in S^1 \times \mathbb{R}.
\end{equation*}
\notag
$$
Its direction field has obvious discrete symmetries, namely, the reflections in the coordinate axes and in the origin
$$
\begin{equation*}
\begin{gathered} \, \varepsilon^1\colon(\theta, c) \to (\theta,-c), \\ \varepsilon^2\colon(\theta, c) \to (-\theta, c), \end{gathered}
\end{equation*}
\notag
$$
and These reflections generate the dihedral group
$$
\begin{equation*}
D_2=\{\operatorname{Id},\varepsilon^1,\varepsilon^2,\varepsilon^3\}= \mathbb{Z}_2 \times \mathbb{Z}_2.
\end{equation*}
\notag
$$
The actions of reflections extend in a natural way to Euler elasticae $(x_t,y_t)$, so modulo the rotations of the $(x,y)$-plane,
$$
\begin{equation*}
\varepsilon^i\colon C \times\mathbb{R}_+\to C \times \mathbb{R}_+,\qquad i=1, 2, 3,
\end{equation*}
\notag
$$
and the image of this map: so that
$$
\begin{equation*}
\varepsilon^i \circ \operatorname{Exp}(\lambda, t)= \operatorname{Exp} \circ \varepsilon^i(\lambda,t), \qquad (\lambda, t) \in C \times \mathbb{R}_+, \quad i=1, 2, 3.
\end{equation*}
\notag
$$
More explicitly,
$$
\begin{equation*}
\begin{aligned} \, \varepsilon^1 \colon (\theta, c, \alpha, \beta, t) &\mapsto (\theta^1, c^1, \alpha, \beta^1, t)=(\theta_t, -c_t, \alpha, \beta, t), \\ \varepsilon^2 \colon (\theta, c, \alpha, \beta, t) &\mapsto (\theta^2, c^2, \alpha, \beta^2, t)=(-\theta_t, c_t, \alpha, -\beta, t), \\ \varepsilon^3 \colon (\theta, c, \alpha, \beta, t) &\mapsto (\theta^3, c^3, \alpha, \beta^3, t)=(-\theta, -c, \alpha, -\beta, t) \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \varepsilon^1\colon (x, y, z, v, w) &\mapsto (x, y, -z, v-xz, w-yz), \\ \varepsilon^2\colon (x, y, z, v, w) &\mapsto (x, -y, z, -v+xz, w-yz), \\ \varepsilon^3\colon (x, y, z, v, w) &\mapsto (x, -y, -z, -v, w). \end{aligned}
\end{equation*}
\notag
$$
The symmetry group $\operatorname{Sym}$ of the exponential map consists of rotations, reflections, and their compositions:
$$
\begin{equation*}
\begin{aligned} \, e^{s \vec h_0},e^{s\vec h_0} \circ \varepsilon^i&\colon C \times \mathbb{R}_+\to C \times \mathbb{R}_+, \\ e^{s X_0},e^{sX_0} \circ \varepsilon^i&\colon G\to G. \end{aligned}
\end{equation*}
\notag
$$
Theorem 2.58. Let $\lambda \in C$. Then for almost all geodesics $\operatorname{Exp}(\lambda,t)$ the first Maxwell time corresponding to the group $\operatorname{Sym}$ of symmetries of the exponential map has the following expression: Remark 2.6. On those geodesics where the first Maxwell time corresponding to the group $\operatorname{Sym}$ is distinct from $t_{\rm Max}^1$, it is greater than this quantity, while $t_{\rm Max}^1$ is the first conjugate time. Theorem 2.59. The function $t_{\rm Max}^1\colon C \to (0,+\infty]$ has the following invariance properties: (1) $t_{\rm Max}^1(\lambda)$ depends only on $E$ and $|\alpha|$; (2) $t_{\rm Max}^1(\lambda)$ is a first integral of the field $\vec H_v$; (3) $t_{\rm Max}^1(\lambda)$ is rotation invariant: if $(\lambda,t) \in C \times \mathbb{R}_+$ and $(\lambda^i, t)=\varepsilon^i(\lambda,t)$, then (4) $t_{\rm Max}^1(\lambda)$ is dilation homogeneous: if $\lambda \in C$ and $\lambda_s=\delta_s(\lambda) \in C$, then 2.10.5. A lower bound for the conjugate time2.10.6. The cut time and length minimizersTheorem 2.62. Let $g_1=(x_1,y_1,z_1,v_1,w_1) \in G$. If $z_1\ne 0$ and $x_1 v_1+y_1 w_1-{(x_1^2+y_1^2)z_1}/{2}\ne 0$, then there exists a unique length minimizer connecting $g_0=\operatorname{Id}$ with $g_1$. 2.10.7. Metric straight linesTheorem 2.63. The following geodesics (and only they) are metric straight lines with natural parametrization on the Cartan group: (1) the one-parameter groups tangent to the distribution:
$$
\begin{equation}
\begin{gathered} \, e^{t(u_1 X_1+u_2 X_2) }=\operatorname{Exp}(\lambda,t), \\ u_1=\cos \theta, \quad u_2=\sin \theta, \quad \lambda=(\theta, c=0, \alpha, \beta)\in C_4\cup C_5\cup C_7, \nonumber \end{gathered}
\end{equation}
\tag{2.131}
$$
(2) the critical geodesics Remark 2.7. Geodesics (2.131) project onto the $(x,y)$-plane as Euclidean straight lines, and the geodesics (2.132) project as critical Euler elasticae (see Fig. 24), called Euler solitons. 2.10.8. Bibliographic commentsThe sub-Riemannian problem on the Cartan group was considered for the first time by Brockett and Dai [56], who showed that geodesics are integrated by elliptic functions. Subsections 2.10.1 and 2.10.3 are based on [123]; § 2.10.2 is based on [124]; § 2.10.4 on [125]–[127]; § 2.10.5 on [141]; §§ 2.10.6 and 2.10.7 on [14]. Also see the recent paper [144]. 3. Instead of a conclusion: several questions left outFor reasons of space we have not addressed several questions close to those considered above. We list them here. 1. Works on vaconomic mechanics [28]. In particular, authors have considered the ‘Chaplygin sledge’ problem in the variational form, and its hydrodynamic analogue, a planar inertial motion of a rigid body in a fluid; this problem has been integrated by elliptic functions [95]. If one of the added masses is let go to infinity, then we obtain a sub-Riemannian problem in the limit [92]. The problem of ball rolling over a plane was integrated by elliptic functions in [90]. In mechanics the sub-Riemannian problem on the motion group of three-dimensional space corresponds to problems in the dynamics of a rigid body with fixed point; it was considered in [93] together with Poinsot’s geometric picture. 2. Left-invariant sub-Finsler problems [16], [17], [24], [25], [30], [37], [38], [41], [42], [52], [57], [102], [138], [139]. 3. Left-invariant sub-Lorentzian problems [26], [61], [80], [82], [91], [147]. 4. Left-invariant sub-Riemannian problems with non-integrable geodesic flow [50], [77], [103], [104], [146]. 5. Applications of left-invariant problems to nilpotent approximation and to a constructive solution of a two-point control problem [4], [7], [31]–[33], [47], [63], [71], [81], [85], [94], [97], [109], [118], [150], [152], [155], [156], [160]. 6. Applications of left-invariant problems to image processing and vision models [34]–[36], [53], [54], [64], [66]–[68], [72]–[74], [110], [112], [113], [120], [121]. 7. Applications of left-invariant problems to robotics [11], [13], [15], [27]. The author is thankful to A. A. Agrachev, V. V. Kozlov, A. V. Podobryaev, A. P. Mashtakov, A. A. Ardentov, and I. Yu. Beschastnyi for their helpful advice on the contents and presentation of this paper. The author is also grateful to E. F. Sachkova for her help in typesetting the text survey and her permanent support during the work on this survey.
Citation in format AMSBIB
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