| Izvestiya: Mathematics |
|
|
|
|
|
On rotation invariant integrable systems A. V. Tsiganov Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2024, Volume 88, Issue 2, DOI: https://doi.org/10.4213/im9506 
Bibliographic databases:
    § 1. IntroductionThe problem of integrability of the Newton equations in the Euclidean space $\mathbb E^n$ is directly related to that of integrability of the Hamiltonian on the phase space $T^*\mathbb E^n$
$$
\begin{equation}
H=\sum_{i=1}^n p_i^2+V(q_1,\dots,q_n),\qquad F_i=-2\,\frac{\partial}{\partial q_i}V(q_1,\dots,q_n)
\end{equation}
\tag{1.2}
$$
and hence, to the possibility of finding $n$ independent invariants for the Hamilton equations
$$
\begin{equation}
\dot{q}_i=\frac{\partial H}{\partial p_i},\quad \dot{p}=-\frac{\partial H}{\partial q_i},\qquad i=1,\dots,n.
\end{equation}
\tag{1.3}
$$
Such invariants include, in particular, functions on the phase space (the first integrals), vector fields (fields of symmetries), differential $k$-forms (which generate $k$-dimensional integral invariants), and so on. For the necessary mathematical definitions and facts, we refer to Kozlov [1]–[3]. The first integrals related by some relations can be found by various methods. Let us mention some known examples. The equation in which the Hamiltonian $H$ (see (1.2)) and some ansatz for the desired first integral $G(q,p)$ are substituted, can be attacked by the brute force method, in which all possible candidates for the solution are tested (see [4] and [5]). Here, $\{\,{\cdot}\,,{\cdot}\,\}$ are the Poisson brackets on the phase space $T^*\mathbb E^n$
$$
\begin{equation}
\{q_i,q_j\}=0,\qquad \{p_i,p_j\}=0,\qquad \{q_i,p_j\}=\delta_{ij}.
\end{equation}
\tag{1.4}
$$
As a preliminary step for the choice of an ansatz for $V(q)$, we can find necessary integration conditions for the potential $V(q)$ using the Kovalewski–Painlevé method and its various generalizations (see [6] and [7]). For example, one can use here the Morales–Ramis method based on analysis of the differential Galois group of the corresponding variational equations along with some partial solutions (see the survey [8]). The use of symmetry fields of the configuration space. Here, in place of the motion equations, we consider the symmetry fields $X_k$, or the Killing vector fields, which preserve the metric of the configuration space. If the potential is invariant under some field of symmetries, then by the Noether theorem there exists a conservation law corresponding to this symmetry (for the corresponding constructions, see [9]). Here, $\mathcal L$ is the Lie derivative along the vector field $X_k$.The use of hidden symmetries of the configuration space. The symmetric products of the symmetry fields $X_j$ of the Euclidean space generate the family of Killing tensors of various valences
$$
\begin{equation*}
K=\sum a_{ij\dots m} X_i\cdot X_j \cdots X_m,\qquad a_{ij\dots m}\in \mathbb R,
\end{equation*}
\notag
$$
which can be used to find integrals of motion which are polynomials of various orders in the momenta (see [10] and [11]). The use of separation variables in the Hamilton–Jacobi equation. Solving for $\alpha_k$ the equations
$$
\begin{equation*}
\Phi_i(u_i,p_{u_i},\alpha_1,\dots, \alpha_n)=0,\qquad \det\biggl[\frac{\partial \Phi_i}{\partial \alpha_j}\biggr]\neq 0
\end{equation*}
\notag
$$
which involve the separation variables $u_i$ and $p_{u_i}$, we obtain $n$ independent functions in the phase space in involution with respect to several Poisson brackets consistent with each other. By imposing some additional conditions on the functions $\Phi_i$, we can find not only the first integrals, but also determine the corresponding action–angle variables (see [12]). If both the separated variables and separated equations are not known, but we know the compatible Poisson brackets in the phase space, we can construct the required number of independent functions in involution $G_k$ using the recursion operator $N$, In this case, $P$ is the Poisson bivector corresponding to the Poisson brackets (see (1.4)), and $P'$ is the Poisson bivector compatible with $P$ (see [13]). The case of degenerate Poisson structures is discussed in [14]. In some cases, the Newton equations (1.1) can be identified with one of the stationary flows of the known non-linear evolution equations (see [15], [16] and also the survey [17]). This allows us to construct the set of first integrals in the finite- dimensional case using the known invariants of non-linear evolution equations. In some cases, the Hamilton equations (1.3) can be written as the Lax equation The sought-for first integrals of motion are generated by the spectral invariants of the Lax matrix $L$, and we need to prove that the number of spectral invariants is sufficient for Liouville integrability of the original Hamiltonian system of equations [18].There are also other methods for constructing the first integrals (see [19] and [20]). For example, to find polynomial and rational integrals of motion for superintegrable systems, one can use the classical theorems by Euler [21], [22], Richelo [23], Chebyshev [24], Riemann–Roch [25], etc. Despite the abundance of classical and new methods for construction of first integrals, the most productive methods for the Euclidean space are the brute force method and the methods using the symmetries of the Euclidean space (see [5]). Applicability of the first method is limited by the computational capabilities for large dimensions of the phase space. In fact, this method works only for two- dimensional integrable systems. The second method also suffers from the dimensionality constraints (for the space of Killing tensors, rather than for the phase space). In fact, this method is most efficient for constructing integrals of motion of first and second order in the momenta (see the discussion in [26]). In this paper, we discuss one possible combination of these two methods, which is capable of partially eliminating the above drawbacks. The computational algorithm is quite simple: $\bullet$ take $m$ fields of symmetries $X_i$ of the configuration space $\mathbb E^n$ that generate $n-2$ independent functions in involution; $\bullet$ determine the coordinates $Q_1$, $Q_2$ and $P_1$, $P_2$ on the phase space $T^*\mathbb E^n$ which are invariant under the above fields $X_i$; $\bullet$ assume that the Hamiltonian $H$ (see (1.2)) and the desired integral of motion $G$ are functions of the invariant variables $Q_1$, $Q_2$, and $P_1$, $P_2$; $\bullet$ solve the equation $\{H(Q,P),G(Q,P)\}=0$ using various substitutions for the function $G$. As a result, we obtain the Hamiltonian $H$ that commutes with $n-2$ linear and quadratic in momenta integrals of motion and with the integral of motion $G$; moreover, this Hamiltonian can be a polynomial of an arbitrary degree or even a rational function with respect to the momenta. We also show how the proposed algorithm works in the case of rotations of the Euclidean space and give some examples of integrable and superintegrable systems which we construct here. As a generalization of the proposed algorithm, we can also use, in place of the Noether symmetry fields, the hidden symmetries and the corresponding integrals of motion. § 2. Rotation invariant integrable systemsThe Euclidean group $\mathrm E(n)$ consists of the isometries of the Euclidean space $\mathbb E^n$, that is, of the transformations of this space which preserve the Euclidean distance between each two points. The direct Euclidean isometries that preserve the chirality of figures form a subgroup of the special Euclidean group $\mathrm{SE}(n)$, whose elements are called rigid motions or Euclidean motions. These elements consist of arbitrary combinations of motions and rotations, but not of reflections, that is, $\mathrm{SE}(n)$ is a semidirect product of the special orthogonal rotation group $\mathrm{SO} (n)$ and the translation group $\mathrm T (n)$. The shifts of $X_i$ along the coordinate axes and rotations of $X_{ij}$ form a basis for $\mathrm{SE}(n)$,
$$
\begin{equation}
X_i=\partial _i, \quad X_{ij}=q_iX_j-q_jX_i,\qquad \partial_k=\frac{\partial}{\partial q_k}.
\end{equation}
\tag{2.1}
$$
In what follows, we will consider the Hamiltonians $H$ (see (1.2)) which are invariant under the action of commutative and non-commutative subgroups of the rotation group $\mathrm{SO}(n)$. The isometries preserve the metric, and hence the invariance requirement for the Hamiltonian $H$ (see (1.2)) is equivalent to that for the potential. Suppose that the potential $V(q_1,\dots,q_n)$ is invariant under some rotation, that is, where $\mathcal L$ is the Lie derivative along the vector field
$$
\begin{equation}
Y_\alpha=\sum c^{ij}_\alpha X_{ij},\qquad c_\alpha^{ij}\in \mathbb R.
\end{equation}
\tag{2.3}
$$
By the Noether theorem, the function
$$
\begin{equation}
M_\alpha=\sum c_\alpha^{ij} J_{ij}, \qquad J_{ik}=q_ip_k-q_kp_i,
\end{equation}
\tag{2.4}
$$
which corresponds to the symmetry field $Y_\alpha$, is in involution with the Hamiltonian $H$ (see (1.2)) with respect to the Poisson brackets (1.4) on the phase space $T^*\mathbb E^n$. That is, the integrals of motion corresponding to the rotation symmetries are combinations of components of the angular momentum tensor $J$. Note that these integrals of motion are one of the oldest integrals, which were extensively studied long before the creation of the Noether theory. Suppose that the Hamiltonian $H$ and $m$ functions $M_\alpha$, $\alpha=1,\dots,m$ give rise to $n-1$ independent integrals of motion in involution. In this case, it remains to find one more additional integral of motion $G$ to construct a Liouville integrable system. The construction of this integral of motion will involve rotations invariant variables. 2.1. Rotation invariant variablesLet us consider the standard representation of the rotation group $\mathrm{SO}(n)$ in space $T^*\mathbb E$: According to [27] and [28], instead of the variables $J_{ij}$, we can use the variables
$$
\begin{equation*}
s_+^{(i)}=\frac{p_i^2}{2},\quad s_-^{(i)}=-\frac{q_i^2}{2},\quad s_3^{(i)}=\frac{q_ip_i}{2},\qquad i=1,\dots,n,
\end{equation*}
\notag
$$
satisfying the Lie–Poisson bracket on the direct sum of $n$ algebras $\mathrm{su}^*(2)$ or $\mathrm{so}^*(3)$,
$$
\begin{equation*}
\{s_3^{(i)},s^{(j)}_\pm\}=\pm \delta_{ij}\, s^{(i)}_\pm,\qquad \{s_+^{(i)},s_-^{(j)}\}=2\delta_{ij}\,s_3^{(i)},
\end{equation*}
\notag
$$
where $q_i$ and $p_i$ are variables satisfying the Poisson brackets (1.4) in the space $T^*\mathbb E$. The substitution of variables $J\to s$ allows us to find a link betern the Gaudin magnets theory and the classical angular momentum theory, the classification of quadrics on Riemannian manifolds of constant curvature, and the classification of orthogonal curvilinear coordinates [27], [28]. According to this theory, the total spin vector with components
$$
\begin{equation*}
S_+=\sum_{i=1}^n s_+^{(i)},\qquad S_-=\sum_{i=1}^n s_-^{(i)},\qquad S_3=\sum_{i=1}^n s_3^{(i)}
\end{equation*}
\notag
$$
is invariant under rotations; this also follows from the classical angular momentum theory (see the discussion in [27] and [28]). In other words, we have the known representation of $\mathrm{so}^*(3)$ in the cotangent bundle $T^*\mathbb E^n$, and this representation is invariant under the rotation group $\mathrm{SO}(n)$. The general case of Riemannian manifolds of constant curvature is not considered in the present paper. In what follows, it will be convenient for us to use the functions
$$
\begin{equation*}
Q_1=\sum_{i=1}^n q_i^2 ,\qquad P_1=\sum_{i=1}^n p_i^2,\qquad A=\{P_1,Q_1\}
\end{equation*}
\notag
$$
instead of $S_\pm$ and $S_3$. The Poisson brackets (1.4) between these variables
$$
\begin{equation*}
\{P_1, A\} =8P_1,\qquad \{Q_1, A\}=- 8Q_1,\qquad \{P_1,Q_1\}=A
\end{equation*}
\notag
$$
are equivalent to Lie–Poisson brackets between elements of the Lie algebras $\mathrm{su}^*(2)$ or $\mathrm{so}^*(3)$ up to renormalization. The corresponding Casimir element for these brackets
$$
\begin{equation*}
J^2\equiv\sum_{i>j}^n J_{ij}^2=P_1Q_1-\frac{1}{16}A^2
\end{equation*}
\notag
$$
is the sum of squared components of the angular momentum tensor. Having at our disposal the rotation-invariant representation of $\mathrm{so}^*(3)$, it is natural to raise the question of constructing representations of the algebra
$$
\begin{equation*}
\mathrm{so}^*(4)=\mathrm{so}^*(3)\times \mathrm{so}^*(3)
\end{equation*}
\notag
$$
in the space $T^*\mathbb E^n$. Below, we will discuss one variant of construction of such a representation. Consider some subgroup of $\mathrm{SO}(n)$ consisting of $m$ rotations of $Y_\alpha$, $\alpha=1,\dots,m$. Suppose that, for this subgroup, there is some second independent solution of equations (2.2) in the space of quadratic polynomials
$$
\begin{equation}
V_2(q_1,\dots,q_n)=\sum_{i,j=1}^n V_{ij}q_iq_j,\qquad V_{ij}\in \mathbb R,
\end{equation}
\tag{2.6}
$$
and this solution is different from $V_1(q)=Q_1$. In this case, the three invariant functions
$$
\begin{equation*}
Q_2=V_2(q_1,\dots,q_n),\qquad P_2=V_2(p_1,\dots,p_n),\qquad B=\{P_1,Q_2\}
\end{equation*}
\notag
$$
commute not with all combinations of components of the angular momentum tensor, but only with $m$ functions $M_\alpha$, $\alpha=1,\dots,m$, corresponding to our chosen symmetry fields
$$
\begin{equation*}
\{Q_2,M_\alpha\}=0,\quad\{P_2,M_\alpha\}=0,\quad \{B,M_\alpha\}=0,\qquad \alpha=1,\dots,m.
\end{equation*}
\notag
$$
This can be proved using the fact that canonical transformation of the symplectic manifold $T^*\mathbb E^n$ preserves the Poisson brackets (1.4) and changes the sign in the angular momentum, and, therefore, in the linear combinations of these components $M_\alpha$ (see (2.4)). Since the matrix $V_{ij}$ (2.6) in the definition of the potential $V_2$ is symmetric, we have so it remains to define the Poisson bracket between the variables $Q_2=V_2(q)$ and $P_2=V_2(p)$. This bracket is a linear function in the momenta $p_1,\dots,p_n$ and is invariant under the action of the chosen subgroup of the rotation group, that is, this bracket is a linear combination of the invariant functions $A$ and $B$.In this paper, we restrict ourselves to considering a special case when the following additional condition on solutions of equations (2.2) in the space of quadratic potentials $V_2(q)$ (see (2.6)) is satisfied: this condition can be written as where $\phi$ is the canonical transformation of (2.7). As far as we know, solutions of equations (2.2) and (2.8) have never been used before for constructing representations of Lie–Poisson algebras.Under condition (2.8), the Poisson brackets between the six functions $Q_{1,2}$, $P_{1,2}$, and $A$, $B$ are given by the Poisson bivector
$$
\begin{equation}
\Pi= \begin{pmatrix} 0 & 0 & -A &-B & -8Q_1 & -8Q_2 \\ 0 & 0 & -B & -A& -8Q_2 & -8Q_1\\ A & B & 0 & 0 & 8P_1 & 8P_2 \\ B & A & 0 & 0 & 8P_2 &8P_1 \\ 8Q_1 & 8Q_2 & -8P_1 &-8P_2 & 0 & 0 \\ 8Q_2 & 8Q_1 &-8P_2 & 8P_1 & 0 & 0 \end{pmatrix}.
\end{equation}
\tag{2.9}
$$
The two Casimir elements of this bivector satisfying the condition of $\Pi\,dC=0$ are as follows:
$$
\begin{equation}
C_1=A^2 + B^2 - 16Q_1P_1 - 16Q_2P_2\quad\text{and}\quad C_2=AB - 8Q_1P_2 - 8Q_2P_1.
\end{equation}
\tag{2.10}
$$
The functions $C_1$ and $C_2$ are the second-order polynomials of $m$ functions $M_\alpha$, $\alpha=1,\dots,m$ appearing in equation (2.5). Proposition 1. Bivector $\Pi$ (see (2.9)) defines the Poisson bracket $\{\,{\cdot}\,,{\cdot}\,\}_\Pi$ equivalent to the Lie–Poisson bracket of the complex Lie algebra $\mathrm{so}^*(4)$. For a proof, we introduce the variables
$$
\begin{equation*}
\begin{alignedat}{3} s_1&=\frac{\mathrm i(A - B)}{16},&\quad s_2&=-\mathrm i(P_1 - P_2) +\frac{\mathrm i(Q_1 - Q_2)}{64},&\quad s_3&=-(P_1 - P_2) -\frac{Q_1-Q_2}{64}, \\ t_1&=\frac{\mathrm i(A + B)}{16},&\quad t_2&=\mathrm{i}(P_1 + P_2) - \frac{\mathrm i(Q_1+Q_2)}{64},&\quad t_3&=(P_1+P_2)+\frac{Q_1 + Q_2}{64}, \end{alignedat}
\end{equation*}
\notag
$$
where $\mathrm i=\sqrt{-1}$. The Poisson bracket $\{\,{\cdot}\,,{\cdot}\,\}_\Pi$, and therefore, bracket (1.4), for these variables coincides with the Lie–Poisson bracket of the algebra $\mathrm{so}^*(4)=\mathrm{so}^*(3)\times \mathrm{so}^*(3)$,
$$
\begin{equation*}
\{s_i,s_j\}=\varepsilon_{ijk}s_k,\quad \{t_i,t_j\}=\varepsilon_{ijk}t_k,\quad \{s_i,t_j\}=0,\qquad i,j,k=1,2,3.
\end{equation*}
\notag
$$
Here, $\varepsilon_{ijk}$ is an antisymmetric tensor. Substituting the variables $s_i$ and $t_i$ into the definitions of the integrals of motion for the Fram–Schottky top, the Steklov top, the Kirchhoff system, and other known integrable systems on $\mathrm{so}^*(4)$ (see [18]), we obtain analogues of these integrable systems on the phase space $T^*\mathbb E^n$, which have no physical meaning. Depending on the chosen subgroup of $\mathrm{SO}(n)$ and the representations of its elements by means of single, double, triple, and similar rotations, equations (2.2) and (2.8) can have several solutions, that is, the dimensionality of the solution space of the form (2.6) depends on the choice of a particular subgroup and its realization. A discussion of the relations between the number of solutions and the particular realization of the rotation subgroup is beyond the scope of this paper. In the case of three or more solutions, one may take either a pair of solutions and construct a representation of the Lie–Poisson algebra $\mathrm{su}^*(2)$, or consider more than two solutions which define the Lie–Poisson algebra of greater dimension, for which the brute force method is not so efficient as for the case of two degrees of freedom. Note that there also exist subgroups of the group $\mathrm{SO}(n)$ and their realizations such that equations (2.2) have solutions in the space of polynomials of third, fourth, and higher degrees with respect to the coordinates. In this case, polynomial Poisson brackets arise in place of linear Poisson brackets. In [29], we consider, in the nine-dimensional Euclidean space, one such system associated with the subgroup $\mathrm{SO}(3)\times \mathrm{SO}(3)$ of the rotation group $\mathrm{SO}(9)$ when the elements of $\mathrm{SO}(3)\times \mathrm{SO}(3)$ are realized as three consecutive rotations. 2.2. New integrable systems on $\mathrm{so}^*(4)$We write the Hamiltonian $H$ (see (1.2)) on the $2n$-dimensional phase space $T^*\mathbb E^n$ in terms of the invariant variables
$$
\begin{equation}
H=\sum_{i=1}^n p_i^2+V(q_1,\dots,q_n)=P_1+U(Q_1,Q_2).
\end{equation}
\tag{2.11}
$$
Let us try to find the integrable potentials $U(Q_1,Q_2)$ by the brute force method [5]. Recall that the brute force method belongs to the class of methods for finding a solution by brute-force search of all possible variants, that is, in our case, we take the Hamiltonian $H$ (see (2.11)) and an ansatz for the second integral of motion $G$, so that we can find a solution for the system of partial differential equations arising from the equation $\{H,G\}_\Pi =0$. Proposition 2. Let us substitute $H$ (see (2.11)) and a second degree polynomial in the momenta $p_1,\dots,p_n$ into the equation A general solution of the resulting system of equations depends on two arbitrary functions $f_1$ and $f_2$ For a proof, we solve equation (2.13). This verifies the uniqueness of solution (2.14), which is directly related to additive separation of variables in the Hamilton–Jacobi equations $H=E_1$ and $G=E_2$,
$$
\begin{equation*}
\begin{aligned} \, (P_1+P_2)+2f_1(Q_1+Q_2)&=E_1+E_2, \\ (P_1-P_2)+2f_2(Q_1-Q_2)&=E_1-E_2. \end{aligned}
\end{equation*}
\notag
$$
Now let us take a more complicated ansatz for the integral of motion $G$ corresponding to two of the five known integrable potentials of degree four in the three- dimensional Euclidean space [7]. Proposition 3. Let us substitute $H$ (see (2.11)) and a particular polynomial of fourth order in momenta The above solutions are partial, since there already exist the solutions described earlier in Proposition 2 and the solutions related to these solutions by transformation of invariant coordinates (see below). Other integrable systems were found in the three-dimensional case in [7], [30], [31]; they were generalized to the $n$-dimensional case in [32], [33]. Proposition 4. Let us substitute $H$ (see (2.11)) and a particular polynomial of fourth order in the momenta In the case of an arbitrary potential $U(Q_1,Q_2)$ in (2.11) and if $G$ is a fourth- degree polynomial of the general form, we have not succeeded in finding either a general solution or other partial solutions of the system of partial differential equations obtained from equation (2.13). Since the solutions of equations (2.2) and (2.8) are defined up to the sign, all the above results also hold after the transformation of the invariant variables That is, the solutions given in Propositions 3 and 4 are partial in the sense that there are also other solutions related to them by transformation (2.20).The first family of integrable potentials $U_{\mathrm{I}}$ (see (2.16)) is invariant with respect to the change of sign. By changing the sign of $Q_2$ in the second (2.17) and the third (2.19) potentials, we obtain two more families of integrable potentials
$$
\begin{equation}
U_{\mathrm{IV}} =a_1(5Q_1 - 3Q_2)(3Q_1 - Q_2)+a_2(5Q_1 - 3Q_2) \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad+\frac{a_3}{Q_1 - Q_2} +\frac{a_4}{Q_1+ Q_2}+ \frac{a_5}{(Q_1 + Q_2)^3},
\end{equation}
\tag{2.21}
$$
$$
\begin{equation}
U_{\mathrm{V}} =a_1(29Q_1^2 + 30Q_1Q_2 + 5Q_2^2) + a_2(5Q_1 + 3Q_2) \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad+\frac{a_3}{Q_1 + Q_2} + \frac{a_4}{Q_1 - Q_2} +\frac{a_5(5Q_1 + 3Q_2)}{(Q_1 - Q_2)^3}.
\end{equation}
\tag{2.22}
$$
Thus, we have five families of integrable systems associated with different subgroups of the rotation group of the $n$-dimensional Euclidean space $\mathbb E^n$. For $a_2=a_3=a_4=a_5=0$, the Hamiltonians $H_N$, $N=\mathrm{I}, \mathrm{II}, \mathrm{III}, \mathrm{IV}, \mathrm{V}$, are the polynomials
$$
\begin{equation*}
H_N=\sum_{i=1}^n p_i^2+\sum_{i,j,k,\ell} \mathcal R_{ijk\ell} q_iq_jq_kq_\ell, \qquad \mathcal R_{ijk\ell}\in \mathbb R,
\end{equation*}
\notag
$$
as well as the corresponding Newton equations
$$
\begin{equation*}
\ddot{q}_i=\sum_{j,k,\ell=1}^n \mathcal R_{ijk\ell}q_j q_k q_\ell.
\end{equation*}
\notag
$$
The fourth-order tensors $\mathcal R$ involved in these equations can sometimes be identified with curvature tensors or Riemann tensors of manifolds seemingly unrelated to the Euclidean space. Examples of integrable systems for which $\mathcal R$ coincides with the constant Riemann tensors on Hermitian symmetric spaces can be found in [15], [16], and [29]. Examples for all of these families of potentials will be given below, and for brevity, we will omit explicit expressions for the integrals of motion $G_N$, $N=\mathrm{I}, \mathrm{II}, \mathrm{III}, \mathrm{IV}, \mathrm{V}$, which are polynomials of fourth order in the momenta $p_1,\dots,p_n$. § 3. Examples for the case $m=n-2$, commutative subgroups of the rotation groupFor $m=n-2$, the above symmetry fields $Y_\alpha$, $\alpha=1,\dots,m$, commute with each other, and give rise to the linear conservation laws $M_\alpha$ (see (2.4)) in involution with respect to the Poisson brackets (1.4). 3.1. Three-dimensional Euclidean space $\mathbb E^3$Consider a rotation about the third axis $X_{12}$ (2.1) in the three-dimensional Euclidean space $\mathbb E^3$. This rotation generates the linear integral of motion Equations (2.2) and (2.8) have two independent solutions
$$
\begin{equation*}
V_1=q_1^2+q_2^2+q_3^2\quad\text{and}\quad V_2=q_1^2+q_2^2 - q_3^2.
\end{equation*}
\notag
$$
Substituting the variables
$$
\begin{equation}
Q_{1,2}=V_{1,2}(q),\qquad P_{1,2}=V_{1,2}(p),\qquad A=\{P_1,Q_1\},\qquad B=\{P_1,Q_2\}
\end{equation}
\tag{3.1}
$$
into potentials (2.16), (2.17), and (2.19), we obtain Liouville integrable systems, for which the Hamilton functions have the form
$$
\begin{equation*}
\begin{aligned} \, H_{\mathrm{I}} &=\sum_{i=1}^3 p_i^2+a_1(q_1^4 + 2q_1^2q_2^2 + 6q_1^2q_3^2 + q_2^4 + 6q_2^2q_3^2 + q_3^4) \\ &\qquad+ a_2(q_1^2 + q_2^2 + q_3^2)+\frac{a_3-a_4}{4(q_1^2 + q_2^2)} +\frac{a_3+a_4}{4q_3^2}, \\ H_{\mathrm{II}} &=\sum_{i=1}^3 p_i^2+4a_1(8q_1^4 + 16q_1^2q_2^2 + 6q_1^2q_3^2 + 8q_2^4 + 6q_2^2q_3^2 + q_3^4) \\ &\qquad+ 2a_2(4q_1^2 + 4q_2^2 +q_3^2)+\frac{a_3}{2(q_1^2 + q_2^2)} + \frac{a_4}{2q_3^2} + \frac{a_5}{8q_3^6}, \\ H_{\mathrm{III}} &=\sum_{i=1}^3 p_i^2+4a_1(q_1^4 + 2q_1^2q_2^2 + 12q_1^2q_3^2 + q_2^4 + 12q_2^2q_3^2 + 16q_3^4) \\ &\qquad+a_2(q_1^2 + q_2^2 + 4q_3^2) +\frac{a_3}{2q_3^2} + \frac{a_4}{2(q_1^2 + q_2^2)} +\frac{a_5(q_1^2 + q_2^2 + 4q_3^2)}{4(q_1^2 + q_2^2)^3}. \end{aligned}
\end{equation*}
\notag
$$
The integrable Hamiltonians with potentials (2.21) and (2.22) are as follows:
$$
\begin{equation*}
\begin{aligned} \, H_{\mathrm{IV}} &=\sum_{i=1}^3 p_i^2+4a_1(q_1^4 + 2 q_1^2 q_2^2 + 6 q_1^2 q_3^2 + q_2^4 + 6 q_2^2 q_3^2 + 8 q_3^4) \\ &\qquad+a_2 (2 q_1^2 + 2 q_2^2 + 8 q_3^2) +\frac{a_3}{2 q_3^2} +\frac{a_4}{2(q_1^2 + q_2^2)} + \frac{a_5}{(2q_1^2 + 2 q_2^2)^3} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation}
\begin{aligned} \, H_{\mathrm{V}} &=\sum_{i=1}^3 p_i^2+ 4a_1(16q_1^4 + 32q_1^2q_2^2 + 12q_1^2q_3^2 + 16q_2^4 + 12q_2^2q_3^2 + q_3^4) \nonumber \\ &\qquad+ 2a_2(4q_1^2 + 4q_2^2 + q_3^2)+ \frac{a_3}{2(q_1^2 + q_2^2)} + \frac{a_4}{2q_3^2} + \frac{a_5(4q_1^2 + 4q_2^2 + q_3^2)}{4q_3^6}. \end{aligned}
\end{equation}
\tag{3.2}
$$
The Casimir elements $C_{1,2}$ (see (2.10)) for this realization (see (3.1)) of the algebra $\mathrm{so}^*(4)$ have the form The above Hamiltonians $H_N$, $N=\mathrm{I},\mathrm{II},\mathrm{III},\mathrm{IV},\mathrm{V}$, commute with $M_1$ and $G_N$,
$$
\begin{equation*}
\{H_N,M_1\}=0,\qquad \{H_N,G_N\}=0,\qquad \{G_N,M_1\}=0,
\end{equation*}
\notag
$$
and so, we have five integrable systems with three degrees of freedom. The polynomial terms in these five potentials were obtained in [7] using the Yoshida method of [6]. The rational (or singular) terms in these Hamiltonians have also been found in the literature; for example, the Hamiltonian $H_{\mathrm{V}}$ at $a_1=a_2=a_3=a_4=0$ was found in [30], [31] as an example of a non-trivial superintegrable system. Indeed, the Hamiltonian $H$ (see (3.2)) with $a_1=a_2=a_3=a_4=0$
$$
\begin{equation*}
H_{\mathrm{V}}=p_1^2+p_2^2+p_3^2+ \frac{a_5(4q_1^2 + 4q_2^2 + q_3^2)}{4q_3^6}
\end{equation*}
\notag
$$
is invariant under the above rotation $X_{12}$ and commutes with the polynomial $G_{\mathrm{V}}$ of order four in the momenta. According to [32] and [33], this Hamiltonian also commutes with polynomials of the second order in the momenta
$$
\begin{equation}
\begin{aligned} \, K_1 &=p_3J_{23} - 2p_1J_{12} + \frac{2a_5q_2(2q_1^2 + 2q_2^2 + q_3^2)}{q_3^6}, \\ K_2 &=p_3J_{13} + 2p_2J_{12} + \frac{2a_5q_1(2q_1^2 + 2q_2^2 + q_3^2)}{q_3^6}, \end{aligned}
\end{equation}
\tag{3.3}
$$
which are not invariant under the rotation $X_{12}$. The algebra of the first integrals has the form
$$
\begin{equation*}
\{M_1,K_1\}=-K_2,\quad \{M_1,K_2\}=K_1,\quad \{K_1,K_2\}=4M_1 {H}_{\mathrm{V}},\qquad M_1=J_{12},
\end{equation*}
\notag
$$
and the fourth-order polynomial
$$
\begin{equation*}
G_{\mathrm{V}}=4M_1^2{H}_{\mathrm{V}} - K_1^2 - K_2^2.
\end{equation*}
\notag
$$
is an element of the centre of this algebra. Since $K_{1,2}$ are not invariant under the rotation $X_{12}$ we cannot find these integrals of motion within our proposed algorithm. A generalization of this superintegrable system to the $n$-dimensional case was given in [32], [33]. Thus, we have completely reproduced the list of polynomial potentials given in Table 1 of [7], except for the first two cases, where $G$ is squared polynomial of second order in the momenta. The cases of integrable potentials absent in this table are considered below. 3.1.1. Double rotationsLet us construct integrable systems invariant under a combination of rotations around the first and third coordinate axes $X_{23}+X_{12}$. The corresponding linear integral of motion is as follows:
$$
\begin{equation*}
M_1=J_{12}+J_{23}=(q_1p_2-p_1q_2) + (q_2p_3 - p_2q_3).
\end{equation*}
\notag
$$
Equations (2.2) and (2.8) have two independent solutions:
$$
\begin{equation*}
V_1=q_1^2+q_2^2+q_3^2\quad\text{and}\quad V_2=2q_1q_3 - q_2^2.
\end{equation*}
\notag
$$
Substituting the corresponding variables $Q_{1,2}$ and $P_{1,2}$ into (2.16) we obtain the integrable Hamiltonian
$$
\begin{equation*}
\begin{aligned} \, H_{\mathrm{I}} &=\sum_{i=1}^3 p_i^2+a_1\bigl(2(q_1^2 + q_2^2 + q_3^2)^2 - (2q_1q_3 - q_2^2)^2\bigr)+a_2(q_1^2 + q_2^2 + q_3^2) \\ &\qquad+\frac{a_3(q_1^2 + q_2^2 + q_3^2)+a_4(2q_1q_3 - q_2^2)} {(q_1 + q_3)^2(q_1^2 - 2q_1q_3 + 2q_2^2 + q_3^2)}. \end{aligned}
\end{equation*}
\notag
$$
Assuming that $a_2=a_3=a_4=0$, we obtain a Hamiltonian associated with the Hermitian symmetric space $\mathrm{Sp}(3)/\mathrm{U}(3)$ of type $\mathrm{C.I}$ (see [15], [16], and [29]). For this partial case, we know also the Lax matrix and the corresponding classical $r$-matrix. However, we do not know any generalization of the Lax matrix to the general case $a_k\neq 0$. For completeness, we give the second and third Hamiltonians (2.17), (2.19) in terms of the original Cartesian coordinates
$$
\begin{equation*}
\begin{aligned} \, H_{\mathrm{II}} &=\sum_{i=1}^3 p_i^2+ a_1(5q_1^2 + 6q_1q_3 + 2q_2^2 + 5q_3^2)(3q_1^2 + 2q_1q_3 + 2q_2^2 + 3q_3^2) \\ &\qquad+a_2(5q_1^2 + 6q_1q_3 + 2q_2^2 + 5q_3^2)+\frac{a_3}{q_1^2 + 2q_1q_3 + q_3^2}+\frac{a_4}{q_1^2 - 2q_1q_3 + 2q_2^2 + q_3^2} \\ &\qquad+\frac{a_5}{(q_1^2 - 2q_1q_3 + 2q_2^2 + q_3^2)^3}, \\ H_{\mathrm{III}} &=\sum_{i=1}^3 p_i^2+a_1(29q_1^4 - 60q_1^3q_3 + 88q_1^2q_2^2 + 78q_1^2q_3^2 - 80q_1q_2^2q_3 - 60q_1q_3^3 \\ &\qquad+ 64q_2^4 + 88q_2^2q_3^2 + 29q_3^4) + a_2(5q_1^2 - 6q_1q_3 + 8q_2^2 + 5q_3^2) \\ &\qquad+\frac{a_3}{q_1^2 - 2q_1q_3 + 2q_2^2 + q_3^2} + \frac{a_4}{(q_1 + q_3)^2} +\frac{a_5(5q_1^2 - 6q_1q_3 + 8q_2^2 + 5q_3^2)}{(q_1 + q_3)^6}. \end{aligned}
\end{equation*}
\notag
$$
The author of the present paper is unaware of any reference in the literature on the integrable Hamiltonians $H_N$, $N=\mathrm{II},\mathrm{III},\mathrm{IV},\mathrm{V}$, 3.2. Four-dimensional Euclidean space $\mathbb E^4$Let us start with construction of integrable systems invariant under the two commuting basis fields of symmetries $X_{12}$ and $ X_{34}$ (see (2.1)). In this case, equations (2.2) and (2.8) have two independent solutions
$$
\begin{equation*}
V_1=q_1^2+q_2^2+q_3^2+q_4^2, \qquad V_2=q_1^2 + q_2^2 - q_3^2 - q_4^2.
\end{equation*}
\notag
$$
Using various realizations of the six-dimensional Lie–Poisson algebra $\mathrm{so}^*(4)$ with $Q_2=\pm V_2(q)$ and expressions for the motion integrals $H_N$ and $G_N$, where $N=\mathrm{I},\mathrm{II},\mathrm{III},\mathrm{IV},\mathrm{V}$, we obtain five integrable systems in the Euclidean space $\mathbb E^4$. Let us now turn to combinations of basis rotations $X_{ij}$ (see (2.1)). In the Euclidean space $E^4$, we can distinguish two types of double rotations. A double rotation is called non-degenerate if all four indices are different (the Clifford displacements)
$$
\begin{equation*}
Y_{ij,km}= X_{ij}+X_{km},\qquad i\neq j,\quad k\neq m .
\end{equation*}
\notag
$$
The remaining dual rotations will be referred to as degenerate (or reducible); in this case, $i=k$ or $i=m$ or $j=k$ or $j=m$. 3.2.1. Non-degenerate double rotationsAs an example, consider the symmetry fields corresponding to sequences of two rotations
$$
\begin{equation*}
Y_{12,34}=X_{12}+X_{34},\qquad Y_{31,42}=X_{31}+X_{42}.
\end{equation*}
\notag
$$
The corresponding Noether integrals
$$
\begin{equation}
M_1= J_{1 2} + J_{3 4},\quad M_2=J_{3 1}+ J_{42},\qquad \{M_1,M_2\}=0,
\end{equation}
\tag{3.4}
$$
commute with each other with respect to the Poisson brackets (1.4). Equations (2.2) and (2.8) have two independent solutions
$$
\begin{equation*}
V_1=\sum_{i=1}^4 q_i^2,\qquad V_2=2(q_1q_4 - q_2q_3).
\end{equation*}
\notag
$$
The Hamiltonian (2.16) with $a_2=a_3=a_4=0$ reads as
$$
\begin{equation*}
H_{\mathrm{I}}=\sum_{i=1}^4 p_i^2+ a_1\biggl(2\biggl(\sum_{i=1}^4 q_i^2 \biggr)^2- (q_1q_4 - q_2q_3)^2\biggr).
\end{equation*}
\notag
$$
The tensor $\mathcal R$ in the definition of the potential coincides with the curvature tensor in the Hermitian symmetric space $\mathrm{SO}(4)/\mathrm{S}(\mathrm{U}(2)\times \mathrm{U}(2))$ of type $\mathrm{A.III}$ (see [15], [16], and [29]). For $a_2=a_3=a_4=a_5=0$, the second and third Hamiltonians (2.17), (2.19) have the form
$$
\begin{equation*}
\begin{aligned} \, H_{\mathrm{II}} &=\sum_{i=1}^4 p_i^2+ a_1(5q_1^2 + 6q_1q_4 + 5q_2^2 - 6q_2q_3 + 5q_3^2 + 5q_4^2) \\ &\qquad\times(3q_1^2 + 2q_1q_4 + 3q_2^2 - 2q_2q_3 + 3q_3^2 + 3q_4^2), \\ H_{\mathrm{III}} &=\sum_{i=1}^4 p_i^2+a_1(29 q_1^4 - 60 q_1^3 q_4 + 58 q_1^2 q_2^2 + 60 q_1^2 q_2 q_3 + 58 q_1^2 q_3^2 + 78 q_1^2 q_4^2 \\ &\qquad - 60 q_1 q_2^2 q_4 - 40 q_1 q_2 q_3 q_4 - 60 q_1 q_3^2 q_4 - 60 q_1 q_4^3 + 29 q_2^4 + 60 q_2^3 q_3 \\ &\qquad + 78 q_2^2 q_3^2 + 58 q_2^2 q_4^2 + 60 q_2 q_3^3 + 60 q_2 q_3 q_4^2 + 29 q_3^4 + 58 q_3^2 q_4^2 + 29 q_4^4). \end{aligned}
\end{equation*}
\notag
$$
The integrable Hamiltonians $H_N$, $N=\mathrm{II}, \mathrm{III}, \mathrm{IV}, \mathrm{V}$, are not related to any Hermitian symmetric space [16]. 3.2.2. Degenerate double rotationsAs an example, consider the symmetry fields corresponding to the double rotations
$$
\begin{equation*}
Y_{12,23}=X_{12}+X_{23},\qquad Y_{14,34}=X_{14}+X_{34}.
\end{equation*}
\notag
$$
The corresponding Noether integrals
$$
\begin{equation}
M_1= J_{1 2} + J_{2 3},\quad M_2=J_{14}+ J_{34},\qquad \{M_1,M_2\}=0,
\end{equation}
\tag{3.5}
$$
commute with each other with respect to the Poisson brackets (1.4). Equations (2.2) and (2.8) have two independent solutions
$$
\begin{equation*}
V_1=\sum_{i=1}^4 q_i^2,\qquad V_2=2q_1q_3 - q_2^2 + q_4^2.
\end{equation*}
\notag
$$
Substituting
$$
\begin{equation*}
Q_{1,2}=V_{1,2}(q),\qquad P_{1,2}=V_{1,2}(p),\qquad A=\{P_1,Q_1\},\qquad B=\{P_1,Q_2\}
\end{equation*}
\notag
$$
into (2.16), (2.17) and (2.19) with $a_2=a_3=a_4=a_5=0$, we obtain
$$
\begin{equation*}
\begin{aligned} \, H_{\mathrm{I}} &=\sum_{i=1}^4 p_i^2+a_1\biggl(2 \biggl(\sum_{i=1}^4 q_i^2\biggr)^2- (2q_1q_3 - q_2^2 + q_4^2)^2\biggr), \\ H_{\mathrm{II}} &=\sum_{i=1}^4 p_i^2+a_1(5q_1^2 + 6q_1q_3 + 2q_2^2 + 5q_3^2 + 8q_4^2)(3q_1^2 + 2q_1q_3 + 2q_2^2 + 3q_3^2 + 4q_4^2), \\ H_{\mathrm{III}} &=\sum_{i=1}^4 p_i^2+a_1(29 q_1^4 - 60 q_1^3 q_3 + 88 q_1^2 q_2^2 + 78 q_1^2 q_3^2 + 28 q_1^2 q_4^2 - 80 q_1 q_2^2 q_3 - 60 q_1 q_3^3 \\ &\qquad- 40 q_1 q_3 q_4^2 + 64 q_2^4 + 88 q_2^2 q_3^2 + 48 q_2^2 q_4^2 + 29 q_3^4 + 28 q_3^2 q_4^2 + 4 q_4^4). \end{aligned}
\end{equation*}
\notag
$$
We do not give explicit expressions for the fourth and fifth integrable cases because they have a similar structure. 3.2.3. Triple rotationsAs an example, consider the symmetry fields corresponding to sequences of the two and three basic rotations
$$
\begin{equation*}
Y_{12,34}=X_{12}+X_{34},\qquad Y_{12,13,24}=X_{12}+X_{13}+X_{24}.
\end{equation*}
\notag
$$
The corresponding Noether integrals
$$
\begin{equation*}
M_1= J_{1 2} + J_{3 4},\quad M_2=J_{1 2} + J_{1 3} + J_{2 4},\qquad \{M_1,M_2\}=0,
\end{equation*}
\notag
$$
commute with each other with respect to the Poisson brackets (1.4). Equations (2.2) and (2.8) have two independent solutions
$$
\begin{equation*}
V_1=\sum_{i=1}^4 q_i^2,\qquad V_2=\frac{\sqrt{5}}{5} (q_1^2 - 4q_1q_4 + q_2^2 + 4q_2q_3 - q_3^2 - q_4^2).
\end{equation*}
\notag
$$
For $a_2=a_3=a_4=0$, the explicit expression for the first Hamiltonian (2.16) is still quite transparent:
$$
\begin{equation*}
H_{\mathrm{I}}=\sum_{i=1}^4 p_i^2+ a_1\biggl(2 \biggl(\sum_{i=1}^4 q_i^2\biggr)^2-\frac{1}{5} (q_1^2 - 4q_1q_4 + q_2^2 + 4q_2q_3 - q_3^2 - q_4^2)^2\biggr).
\end{equation*}
\notag
$$
We do not give explicit expressions for the other four Hamiltonians due to their bulkiness. We also do not discuss the question of equivalence of representations of different subgroups.
§ 4. Examples for the case $m>n-2$For $m>n-2$, the above rotation symmetry fields $Y_\alpha$, $\alpha=1,\dots,m$ may fail to commute with each other, but we will nevertheless require the existence of $n-2$ linear or quadratic conservation laws in involution with respect to the Poisson brackets (1.4). Recall that, in classical mechanics, a Hamiltonian system on a $2n$-dimensional symplectic manifold is called Liouville integrable if there are $n$ functionally independent integrals of motion in involution. A system is superintegrable if it admits more than $n$ integrals which are functionally independent and commute with the Hamiltonian (see the survey [20] and the references therein). If these superintegrable systems are integrable in Abel quadratures, then the integrals of motion are expressed in terms of the action variables and the angle variables [19] using the Euler theorem [21], the Riemann–Roch theorem [25], the Chaplygin theorem [24], and other classical theorems from various areas of mathematics and its applications (see [23] and [22]). Within the framework of our proposed algorithm, we cannot construct additional integrals of motion at $m=n-2$, since these integrals of motion will not be invariant under the chosen symmetry fields (see example (3.3)). If $m>n-2$, we obtain the superintegrable systems, since the symmetry fields do not commute with each other. In what follows, we restrict ourselves only to two examples from the abundant set of such superintegrable systems. 4.1. Five-dimensional Euclidean space $\mathbb E^5$Consider two pairs of commuting double rotations
$$
\begin{equation*}
Y_{12,34}=X_{12}+X_{34},\qquad Y_{31,42}=X_{31}+X_{42}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
Y_{12,23}=X_{12}+X_{23},\qquad Y_{14,34}=X_{14}+X_{34},
\end{equation*}
\notag
$$
which we have used for construction of integrable systems in the four-dimensional Euclidean space. 4.1.1. Non-degenerate double rotationsIn the five-dimensional Euclidean space $\mathbb E^5\!$, there are no double rotations that commute with $Y_{12,34}$ and $Y_{31,42}$. Nevertheless, adding two more linear integrals of motion to the Noether integrals of motion
$$
\begin{equation*}
M_1= J_{1 2} + J_{3 4},\quad M_2=J_{3 1}+ J_{42},\qquad \{M_1,M_2\}=0
\end{equation*}
\notag
$$
(see (2.4), (3.4)), we get the following algebra of integrals:
$$
\begin{equation*}
\{M_1, M_3\}=- M_4,\quad \{M_1, M_4\}= M_3, \quad \{M_2, M_3\}=M_4,\quad \{M_2 M_4\} =- M_3.
\end{equation*}
\notag
$$
From this system we can construct the desired number of $n-2=3$ independent integrals of motion in involution Solutions of equations (2.2) and (2.8), which commute with $m=4$ functions $M_1$, $M_2$, $M_3$, $M_4$, have the form
$$
\begin{equation*}
V_1(q)=\sum_{i=1}^5 q_i^2,\qquad V_2(q)=\pm(2q_1q_4 - 2q_2q_3 + q_5^2).
\end{equation*}
\notag
$$
Substituting the corresponding invariant variables
$$
\begin{equation*}
Q_{1,2}=V_{1,2}(q),\qquad P_{1,2}=V_{1,2}(p),\quad A=\{P_1,Q_1\},\quad B=\{P_1,Q_2\}
\end{equation*}
\notag
$$
into the expressions for $H_N$ and $G_N$, $N=\mathrm{I},\mathrm{II},\mathrm{III},\mathrm{IV},\mathrm{V}$ we obtain two functions commuting with the four functions $M_1$, $M_2$, $M_3$ and $M_4$, that is, we get five superintegrable Hamiltonians, two of which are associated with the Hermite symmetric spaces B.III and BD.I of types [15], [16], [29]. 4.1.2. Degenerate double rotationsLet us add two linear integrals of motion to the Noether integrals
$$
\begin{equation*}
M_1= J_{1 2} + J_{2 3},\quad M_2=J_{14}+ J_{34},\qquad \{M_1,M_2\}=0,
\end{equation*}
\notag
$$
which commute with each other (see (2.4), (3.5)). As a result, we get
$$
\begin{equation*}
\{M_1, M_3\}=0,\quad \{M_1, M_4\}=0,\quad \{M_2, M_3\}= \sqrt{2}\,M_4,\quad \{M_2, M_4\}=-\sqrt{2}\,M_3
\end{equation*}
\notag
$$
and The solutions of equations (2.2) and (2.8) which commute with $m=4$ functions $M_1$, $M_2$, $M_3$ and $M_4$ have the form
$$
\begin{equation*}
V_1(q)=\sum_{i=1}^5 q_i^2,\qquad V_2(q)=\pm(q_1^2 + 2q_2q_3 - q_4^2 - q_5^2).
\end{equation*}
\notag
$$
Substituting the corresponding rotation invariant variables into the expressions for $H_N$, $N=\mathrm{I},\mathrm{II},\mathrm{III},\mathrm{IV},\mathrm{V}$, we obtain five superintegrable Hamiltonians in the five- dimensional Euclidean space $T^*\mathbb E^5$.
§ 5. ConclusionsTo construct integrable and superintegrable systems in the $n$-dimensional Euclidean space, we propose to use $m$ commuting or non-commuting fields of symmetries. If these fields of symmetries are sufficient to construct $n-2$ integrals of motion in involution, then the remaining two integrals of motion can be constructed using variables invariant under these fields of symmetries. In the present paper, this method is used for construction of a number of known and unknown integrable and superintegrable systems associated with various realizations of the Lie algebra $\mathrm{so}^*(4)$. Since there is an infinite number realizations and the corresponding integrable systems, we restrict ourselves to considering only a few examples in Euclidean spaces of dimension $\leqslant 5$. A natural generalization of the proposed algorithm consists of full or partial replacement of isometries by hidden symmetries associated with the second-order Killing tensors. In this case, the Noether integrals of motion $M_\alpha$ (see (2.4)) will be replaced by the quadratic integrals of motion of the form
$$
\begin{equation*}
h_\alpha= \sum_{i,j=1}^m c^{ij}_\alpha M_{ij}^2 +t_\alpha(p)+v_\alpha(q),
\end{equation*}
\notag
$$
where the functions $M_{ij}=J_{ij}$ correspond to the rotations of $X_{ij}$, the functions $t_\alpha(p)$ correspond to the combination of translations $X_i$ (2.1) and $v_\alpha(q)$ is some function of coordinates $q_i$. The corresponding integrable potentials are not solutions of the equation $\mathcal L_{Y_\alpha}V(q)=0$ (see (2.2)), but the solutions of the equations $d( \mathcal K_\alpha \,dV(q))=0$, where $\mathcal K_\alpha$ are the Killing tensors corresponding to the quadratic conservation laws $h_\alpha$. Examples of such integrals of motion $h_\alpha$, the Killing tensors $\mathcal K_\alpha$ with nonzero Haantjes torsion, and corresponding integrable systems can be found in [29].
Citation in format AMSBIB
\Bibitem{Tsi24} |
|
||||||||||||||||||||||||||
|
Contact us: |
Terms of Use
|
Registration to the website |
Logotypes |
|