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On the Poincaré problem of the third integral of the equations of rotation of a heavy asymmetric top V. V. Kozlov Steklov Mathematical Institute of Russian Academy of Sciences, Moscow § 1. The problem of rotation of a rigid body about a fixed pointThe configuration space $G$ of the top rotation problem is the group $\operatorname{SO}(3)$. From the viewpoint of Hamiltonian formalism, the phase space is the space of the cotangent bundle $\Gamma=T^*G=\operatorname{SO}(3)\times \mathbb{R}^3$. The Hamiltonian function $H$ has the form where $T$ is the kinetic energy (the Hamiltonian of the Euler–Poinsot problem), and $V$ is the potential energy due to gravity.The Hamiltonian equations have two integrals: the total energy $H$ and the projection of the kinetic moment of the rigid body to the vertical axis $F$ (the area integral) are preserved. The second integral, being Noetherian, is generated by the one-parameter symmetry group $g$, which is the group of rotations $\operatorname{SO}(2)$ along the vertical axis. The presence of the symmetry group enables one to reduce the number of degrees of freedom to two with due account of the fixed constant of the area integral $F=c$. The phase space of the reduced Hamiltonian system is $T^*S^2$, where $S^2=\operatorname{SO}(3)/\operatorname{SO}(2)$ is the two-dimensional sphere. This reduction of the order was well-known before advent of the general theory of the reduction of sorder for systems with symmetry. The thing is that the rotation of a rigid body can be described by the Euler–Poisson equations
$$
\begin{equation}
I\dot{\omega}+\omega\times I\omega=\mu\gamma\times r,\qquad \dot{\gamma}=\gamma\times \omega,
\end{equation}
\tag{1.2}
$$
where $\omega$ is the angular velocity of the top relative to a moving frame associated with the rigid body, $I$ is the constant inertia operator, $\gamma$ is the unit vector on the vertical axis considered relative to the same moving frame, $\mu$ is the body weight, and $r$ is the radius vector of its centre of mass. If the value of the area integral $F=(I\omega,\gamma)$ is fixed, then (1.2) are equations of the reduced system, but represented in the redundant non-canonical variables. In order to completely describe the rotation of the top in the three-dimensional space, we may add two more equations
$$
\begin{equation}
\dot{\alpha}=\alpha \times \omega,\qquad \dot{\beta}=\beta \times \omega,
\end{equation}
\tag{1.3}
$$
which describe the evolution of the two other unit vectors $\alpha$, $\beta$ of the fixed space in the moving frame. The manifold of the group $\operatorname{SO}(3)$ is defined by the following clear equations in the nine-dimensional space:
$$
\begin{equation*}
(\alpha,\alpha)=(\beta,\beta)=(\gamma,\gamma)=1,\qquad (\alpha,\beta)=(\beta,\gamma)=(\gamma,\alpha)=0.
\end{equation*}
\notag
$$
According to the Liouville complete integrability theorem of Hamiltonian systems for the solution of the top rotation problem, in the gravity field it is sufficient to know one more integral $\Phi\colon \Gamma \to \mathbb{R}$, which would be in involution with the area integral: $\{F,\Phi\}=0$. The problem of existence of a third single-valued analytic integral was posed by Poincaré in § 86 of his celebrated New methods of celestial mechanics [1]. Poincaré discusses the possibility of application of his general theory for integrability of the “general dynamics problem”. Here, we speak about the existence of the third integral in the form of a power series
$$
\begin{equation}
\Phi=\Phi_0+\varepsilon\Phi_1+\varepsilon^2\Phi_2+\cdots
\end{equation}
\tag{1.4}
$$
for the Hamiltonian system, in which Hamiltonian (1.1) is replaced by the function $T+\varepsilon V$. The idea of introduction of the small parameter $\varepsilon$ is clear – this is $\mu|r|$. The coefficients in (1.4) are assumed to be single-valued analytic functions defined on the bundle $\Gamma$. Poincaré’s analysis lead him to unequivocal conclusions. He points out that “… nothing excludes the possible existence of a third single-valued integral…” Nevertheless, from the analysis of the asymmetric case Poincaré concludes that “… this third integral cannot be algebraic in the general case”. The last assertions is nothing else as the famous Poincaré theorem on the absence of additional algebraic integrals in the problem of rotation of a heavy asymmetric rigid body about a fixed point. In fact, this Poincaré’s remark looks rather mysterious. In which variables is this third integral assumed to be algebraic? Are these the action–angle variables of the Euler–Poinsot problem, which was used by Poincaré in the analysis of the problem on integrals of the form (1.4)? But in these variables even the Hamiltonian $H$ (the energy integral) is not an algebraic function. Apparently, Poincaré’s contemporaries did not fully understand his arguments underlying Poincaré’s theorem on algebraic integrals. It may well be that this fact prompted Husson to give another proof of Poincaré’s theorem on an asymmetric heavy top [2]. Husson’s theorem (nowadays called the Poincaré–Husson theorem) deals with equations (1.2), rather than with the canonical Hamiltonian equations, which were analyzed by Poincaré’s. By the way, Husson gives no comments on Poincaré’s approach to proving the non-existence of an additional algebraic integral (we will get back to this question in § 3). The “direct” Husson’s proof on algebraic integrals of equations (1.2) is based on the available conditions for algebraically of abelian integrals on the corresponding Riemannian surfaces. A more significant problem that remains in the shadows is the following. Husson’s theorem deals with algebraic integrals of equations (1.2). However, from the viewpoint of Hamiltonian formalism, all such integrals are in involution with the area integral. Meanwhile, Poincaré posed the problem on the third analytic integral of the form (1.4), which in general does not commute with the area integral. The analysis performed by Poincaré’s was concerned with the complete problem on rotation of a rigid body qua Hamiltonian system with three degrees of freedom. The problem of non-existence of an additional analytic integral of the form (1.4) of the reduced problem on the rotation of an asymmetric rigid body was solved in [3] and [4]. In addition to Poincaré’s ideas given in the fifth chapter of [1], these papers utilize some novel ideas. Suppose that the equations of rotation of the top admit a third analytic integral $\Phi$ of the form (1.4). The question is whether the motion equations have an additional analytic integral $\widehat{\Phi}$ which would be in involution with the area integral $F$. Of course, like $\Phi$, the integral $\widehat{\Phi}$ should be independent of the other two integrals $H$ and $F$. The answer here is most likely negative. Nevertheless, there are some meaningful results in this direction. Recall that a Hamiltonian system with three degrees of freedom describing the motion of the top admits a one-parameter symmetry group $g$, which coincides with the group of rotations of the body about the vertical axis. The action of this group on $\operatorname{SO}(3)$ extends to the action on the phase space $\Gamma=\operatorname{SO}(3)\times \mathbb{R}^3$. The orbits of this group are circles. Theorem 1. Assume that the heavy top problem has an analytic integral $\Phi$, and let $\widehat{\Phi}$ be the result of averaging of $\Phi$ over the orbits of the extended action of $g$ on $\Gamma$. Then $\widehat{\Phi}$ is a first integral of the motion equations, which is in involution with the integral $F$. This theorem does not solve the above problem, because the function $\widehat{\Phi}$ may depend on the other integrals $H$ and $F$, or even be a constant. Let us prove Theorem 1 by using concrete canonical variables in the phase space $\Gamma$ – these being the Euler angles $\vartheta$, $\varphi$, $\psi$ and the conjugate canonical momenta $p_\vartheta$, $p_\varphi$, $p_\psi$. The Hamiltonian function $H$ is independent of the precession angle $\psi$. The action of the group $g$ is reduced to the translation $\psi \mapsto \psi+c$, $c \in \mathbb{R}$, the remaining variables remaining intact. In these variables, $F=p_\psi$, and the function $\Phi$ may depend on all canonical variables. Since $\Phi$ is the first integral, we have
$$
\begin{equation}
\begin{aligned} \, \nonumber \dot{\Phi}&=\frac{\partial\Phi}{\partial\vartheta}\, \frac{\partial H}{\partial p_\vartheta}+ \frac{\partial\Phi}{\partial p_\vartheta} \biggl(-\frac{\partial H}{\partial\vartheta}\biggr)+ \frac{\partial\Phi}{\partial\varphi}\, \frac{\partial H}{\partial p_\varphi}+ \frac{\partial\Phi}{\partial p_\varphi} \biggl(-\frac{\partial H}{\partial\varphi}\biggr) \\ &\qquad+\frac{\partial\Phi}{\partial\psi}\,\frac{\partial H}{\partial p_\psi}+ \frac{\partial\Phi}{\partial p_\psi} \biggl(-\frac{\partial H}{\partial\psi}\biggr)=0. \end{aligned}
\end{equation}
\tag{1.5}
$$
The last term in this formula vanishes, since $H$ is independent of $\psi$. We now set
$$
\begin{equation*}
\widehat{\Phi}(\vartheta,\varphi,p_\vartheta,p_\varphi,p_\psi)= \frac{1}{2\pi}\int_0^{2\pi}\Phi\,d\psi.
\end{equation*}
\notag
$$
It is clear that $\widehat{\Phi}$ (as well as $\Phi$) is an analytic function on $\Gamma$. Averaging now both sides of (1.5) over $\psi$ and since $H$ is independent of $\psi$, we have
$$
\begin{equation*}
(\widehat{\Phi})^{\cdot}=\frac{\partial\widehat{\Phi}}{\partial\vartheta}\, \frac{\partial H}{\partial p_\vartheta}+ \frac{\partial\widehat{\Phi}}{\partial p_\vartheta} \biggl(-\frac{\partial H}{\partial\vartheta}\biggr)+ \frac{\partial\widehat{\Phi}}{\partial\varphi}\, \frac{\partial H}{\partial p_\varphi}+ \frac{\partial\widehat{\Phi}}{\partial p_\varphi} \biggl(-\frac{\partial H}{\partial\varphi}\biggr)=0.
\end{equation*}
\notag
$$
Therefore, $\widehat{\Phi}$ is a first integral independent of $\psi$. In particular, $\{\widehat{\Phi},F\}= 0$, the result required. In the present note, we prove the absence of an additional single-valued analytic integral $\Phi$ of the form (1.4) without assumption that the integrals $\Phi$ and $F$ should be involutive. We first establish some general result on the absence of a third single-valued analytic integral for Hamiltonian system with degenerate unperturbed function (see § 2) – the Hamiltonian equations in the action–angle variables of the Euler–Poinsot problem are precisely of this form. Nevertheless, due to singularities of the Hamiltonian in the action–angle variables, this general result does not directly apply to the above problem of an asymmetric top. This additional difficulty is circumvented in § 3. In § 4, we discuss the problem of an additional integral which is a polynomial in momenta (like the other two integrals $H$ and $F$). § 2. Theorem on the third integralWe first consider an auxiliary problem on integrals of a Hamiltonian system with three degrees of freedom and the Hamiltonian
$$
\begin{equation}
H=H_0(y_1,y_2)+\varepsilon H_1(y_1,y_2,y_3,x_1,x_2).
\end{equation}
\tag{2.1}
$$
Here, $y$, $x\ \operatorname{mod} 2\pi$ are the action–angle variables of the unperturbed system, which is assumed to be non-degenerate in the variables $y_1$, $y_2$, that is,
$$
\begin{equation*}
\begin{vmatrix} \dfrac{\partial^2 H_0}{\partial y_1^2} & \dfrac{\partial^2 H_0}{\partial y_1\,\partial y_2} \\ \dfrac{\partial^2 H_0}{\partial y_1\,\partial y_2} & \dfrac{\partial^2 H_0}{\partial y_2^2} \end{vmatrix}\ne 0,
\end{equation*}
\notag
$$
in some open domain $D$ in the space of $y_1$, $y_2$. We suppose that the phase space of this system is the direct product $\Gamma=B \times \mathbb{T}^3$, where $\mathbb{T}^3=\{x_1,x_2,x_3\ \operatorname{mod}2\pi\}$ is the three-dimensional torus, and $B=D \times I$, where $I$ is some open interval of the real axis $\mathbb{R}=\{y_3\}$. We will also assume that, for all $\varepsilon$, the function $H$ is real analytic in the domain $B \times \mathbb{T}^3$. In the actual fact, $H$ is defined and analytic in the domain $B \times \mathbb{T}^2$, where $\mathbb{T}^2=\{x_1,x_2\ \operatorname{mod}2\pi\}$. The Hamiltonian of the problem of rotation of a heavy body about a fixed point in the action–angle variables of the Euler–Poinsot problem (see [1], [5]) has the same form. This Hamiltonian system has two clear integrals: $H$ and $F=y_3$. Following the general Poincaré problem, we will try to find a third single-valued integral $\Phi(y,x,\varepsilon)$, which would be an analytic function of the seven variables $y$, $x\ \operatorname{mod}2\pi$, and $\varepsilon$. In the actual fact, we will consider an even more general problem of finding such an integral in the form of a formal series
$$
\begin{equation}
\Phi_0(y,x)+\varepsilon\Phi_1(y,x)+\varepsilon^2\Phi_2(y,x)+\cdots
\end{equation}
\tag{2.2}
$$
with analytic coefficients in the domain $B \times \mathbb{T}^3$. Here, the power series (2.2) is not assumed to be convergent. A formal series in powers of $\varepsilon$ is zero if and only if all its coefficients vanish identically. This series will be called an additional non-trivial integral if the following conditions are satisfied: 1) the Poisson bracket vanishes,
$$
\begin{equation*}
\{H,\Phi\}=\{H_0,\Phi_0\}+\varepsilon[\{H_0,\Phi_1\}+ \{H_1,\Phi_0\}]+\cdots\equiv 0;
\end{equation*}
\notag
$$
2) the rank of the Jacobi matrix
$$
\begin{equation}
\begin{Vmatrix} \dfrac{\partial H}{\partial y_1} & \dfrac{\partial H}{\partial y_2} & \dots & \dfrac{\partial H}{\partial x_3} \\ \dfrac{\partial F}{\partial y_1} & \dfrac{\partial F}{\partial y_2} & \dots & \dfrac{\partial F}{\partial x_3} \\ \dfrac{\partial \Phi}{\partial y_1} & \dfrac{\partial \Phi}{\partial y_2} & \dots & \dfrac{\partial \Phi}{\partial x_3} \end{Vmatrix}
\end{equation}
\tag{2.3}
$$
is $3$; in other words, at least one of the minors of order $3$ of this matrix (considered as a formal series in powers of $\varepsilon$) is non-zero. The second row in matrix (2.3) is, of course, $(0,0,1,0,0,0)$. We expand the perturbing function as a multiple Fourier series:
$$
\begin{equation}
H_1=h_0(y_1,y_2,y_3)+\sideset{}{'}\sum_{\substack{m_1,m_2 \in \mathbb{Z}\\ m_1^2+m_2^2 \ne 0}}h_{m_1m_2}(y)e^{i(m_1x_1+m_2x_2)}.
\end{equation}
\tag{2.4}
$$
All coefficients of this series are analytic functions of $y_1$, $y_2$, $y_3$. It is clear that
$$
\begin{equation*}
h_0=\frac{1}{4\pi^2}\int_{\mathbb{T}^2}H_1\,dx_1\,dx_2.
\end{equation*}
\notag
$$
We set These are the frequencies of conditionally periodic motions on two-dimensional invariant tori of the unperturbed problem, which are defined in terms of the variables $y_1$ and $y_2$.Definition 1. A secular set $\mathscr{B}$ of a system with Hamiltonian (2.1) is a set of points $(y_1,y_2) \in D$ satisfying the following conditions: 1) $m_1\omega_1(y)+m_2\omega_2(y)=0$, $m_1,m_2 \in \mathbb{Z}$, 2) $|m_1|+|m_2| \ne 0$, 3) $h_{m_1m_2}(y_1,y_2,y_3) \not\equiv 0$ qua function of $y_3$. By a secular set, we also understand the set of all resonant tori in the phase space of the unperturbed problem corresponding to the action variables $(y_1,y_2) \in \mathscr{B}$. We let $\mathscr{A}(M)$ denote the class of real functions analytic in a connected domain $M \subset \mathbb{R}^n$. Definition 2. A set $\mathscr{N} \subset M$ is said to be a key set for the class $\mathscr{A}(M)$ if $f \equiv 0$ on the domain $M$ for each function $f$ from the class $\mathscr{A}(M)$ which vanishes on $\mathscr{N}$. Let $W$ be a subdomain of the domain $D$ and $\overline{W} \subset D$. Theorem 2. Assume that a system with Hamiltonian (2.1) satisfies the following conditions: 1) the Hessian $|\partial^2 H_0/\partial y^2| $ is non-zero in the domain $D$, 2) $\mathscr{B} \cap W$ is a key set for the class $\mathscr{A}(W)$, 3) the function $H_0$ has no critical points in the domain $D$, 4) $\partial h_0/\partial y_3 \not\equiv 0$ in the domain $B=D\times I$. Then the system with Hamiltonian function (2.1) has no first integral $\Phi(y,x,\varepsilon)$ that is analytic in the domain $\Gamma \times (-\varepsilon_0,\varepsilon_0)$, $\varepsilon_0>0$ and independent the functions $H$ and $F$. Condition 4) is essential. Let us assume that the perturbing function is independent of the action variables $y_3$. Then $x_3$ is an additional integral. However, $x_3$ is multi-valued. But $\sin x_3$ is a single-valued analytic integral. Theorem 2 is proved by the Poincaré method. We first show that the function $\Phi_0$ is independent of the angle variables $x_1$ and $x_2$. Indeed, $\Phi_0$ is a first integral of the unperturbed system. Assume that the two-dimensional torus $y_1=y_1^0$, $y_2=y_2^0$ is non-resonant: the ratio of the frequencies $\omega_1/\omega_2$ is irrational. For all $y_3$, $x_3$, the function $\Phi_0(y_1^0,y_2^0,y_3,x_1,x_2,x_3)$ is independent of $x_1$ and $x_2$, since any phase trajectory is dense on this torus and $\Phi_0$ is constant on the solutions of the unperturbed problem. It remains to use the continuity of the function $\Phi_0$ and the density of the set of non-resonant tori of the unperturbed system satisfying condition 1) of the theorem. Let us now show that the function $\Phi_0$ is independent of $x_3$. To this end, we will employ the equality In the canonical action–angle variables of the unperturbed problem, this equality can be written explicitly as
$$
\begin{equation}
\frac{\partial\Phi_1}{\partial x_1}\,\frac{\partial H_0}{\partial y_1}+ \frac{\partial\Phi_1}{\partial x_2}\,\frac{\partial H_0}{\partial y_2}- \frac{\partial H_1}{\partial x_1}\,\frac{\partial\Phi_0}{\partial y_1}- \frac{\partial H_1}{\partial x_2}\,\frac{\partial\Phi_0}{\partial y_2} +\frac{\partial H_1}{\partial y_3}\, \frac{\partial\Phi_0}{\partial x_3}=0.
\end{equation}
\tag{2.6}
$$
Averaging the left-hand side of this equality over the angle variables $x_1$, $x_2$ and since the functions $H_0$ and $\Phi_0$ are independent of these variables, we get that
$$
\begin{equation*}
\frac{\partial h_0}{\partial y_3}\,\frac{\partial\Phi_0}{\partial x_3}=0.
\end{equation*}
\notag
$$
From condition 4) it follows that $\partial h_0/\partial y_3\not\equiv 0$. The ring of analytic functions has no zero divisors, and hence $\partial\Phi_0/\partial x_3 \equiv 0$. This shows that $\Phi_0$ is independent of $x_3$. Thus, $\Phi_0$ is an analytic function of the variables $y_1$, $y_2$ and $y_3$. Now equality (2.6) can be written as
$$
\begin{equation}
\frac{\partial\Phi_1}{\partial x_1}\, \omega_1+\frac{\partial\Phi_1}{\partial x_2}\, \omega_2- \frac{\partial H_1}{\partial x_1}\,\frac{\partial\Phi_0}{\partial y_1}- \frac{\partial H_1}{\partial x_2}\,\frac{\partial\Phi_0}{\partial x_2}=0.
\end{equation}
\tag{2.7}
$$
This equality is satisfied for all values of the momentum $y_3$, which plays the role of the parameter. We now expand the function $\Phi_1$ as a double Fourier series in the angle coordinates $x_1$ and $x_2$, The coefficients $\varphi_{m_1m_2}$ are analytic functions of $y_1$, $y_2$, $y_3$ and $x_3$. Substituting expansions (2.4) and (2.8) into (2.7) and equating the coefficients of the harmonics $\exp i(m_1x_1+m_2x_2)$, we get
$$
\begin{equation*}
(m_1\omega_1+m_2\omega_2)\varphi_{m_1m_2}- \biggl(m_1\, \frac{\partial\Phi_0}{\partial y_1}+ m_2\,\frac{\partial\Phi_0}{\partial y_2}\biggr)h_{m_1m_2}=0.
\end{equation*}
\notag
$$
Now we choose $y_1$, $y_2$ from the secular set $\mathscr{B}$. In this case, we have
$$
\begin{equation*}
m_1\, \frac{\partial H_0}{\partial y_1}+ m_2\, \frac{\partial H_0}{\partial y_2}=0\quad\text{and}\quad m_1\, \frac{\partial\Phi_0}{\partial y_1}+ m_2\, \frac{\partial\Phi_0}{\partial y_2}=0.
\end{equation*}
\notag
$$
Next, $|m_1|+|m_2| \ne 0$, and hence, for all $y_3$, the Jacobian of the functions $H_0$ and $\Phi_0$ (in the variables $y_1$ and $y_2$) vanishes on the secular set. But then (by the second condition of the theorem) these functions are dependent everywhere in the domain $W$ (and, therefore, in the domain $D$). Now using condition 3) of the theorem (together with conditions 1) and 2)) it can be shown that there exists a sufficiently small convex domain $E \subset W$ ($\overline{E} \subset W$) such that $1^\circ$. $\mathscr{B} \cap E$ is a key set for $\mathscr{A}(E)$, $2^\circ$. $\Phi_0=\mathscr{R}(H_0,F)$ for all $y_3 \in I'$ ($I'$ is some subinterval of $I$) in the domain $E$, where $\mathscr{R}$ is a function analytic in the first argument in the interval $(\delta',\delta'')$, $\delta'=\min_E H_0$, $\delta''=\max_E H_0$ ($\delta'<\delta''$), $3^\circ$. the function is a first integral of the Hamiltonian system with Hamiltonian (2.1) and is analytic in the direct product
$$
\begin{equation*}
E \times I' \times \mathbb{T}^3 \times (-\varepsilon',\varepsilon'),
\end{equation*}
\notag
$$
where $\varepsilon'$ is some positive constant. This result is proved in the same way as Lemmas 2 and 3 in [5], Chap. 1, § 1. To complete the proof of Theorem 2, we will use the Poincaré method. Let be the third integral of the Hamiltonian equations. If the functions $H$, $F$ and $\Phi$ are independent, then the rank of the Jacobi matrix (2.3) is 3. Let $J(y,x,\varepsilon)$ be a non-identically zero minor of order 3 of this matrix. We expand the analytic function $J$ as a series in powers of $\varepsilon$, As we have just proved, $J_0 \equiv 0$. Assume that expansion (2.10) starts with the term $\varepsilon^p J_p$, $J_p \not\equiv 0$.Function (2.9) is an analytic integral, which we denote by $\Phi'(y,x,\varepsilon)$. We have Applying now the above operation to the integral $\Phi'$, we find that $J_0=J_1 \equiv 0$ in (2.10). Repeating this operation $p$ times, we obtain a contradiction to the assumption $J_p \not\equiv 0$. This completes the proof of Theorem 2.Remarks. $1^\circ$. The proof shows that the variable $x_3$ is not necessarily an angle variablle. $2^\circ$. Under the assumptions of Theorem 2, there is even no an additional formal integral of the form (2.2) with single-valued analytic coefficients. $3^\circ$. Condition 4) of Theorem 2 can be replaced by the following more general one: for some integer $k$, For $k=0$, we get the available condition if $m_1$, $m_2$ are such that $h_{m_1,m_2}\not\equiv 0$.§ 3. An asymmetric top theoremTheorem 3. If a rigid body is dynamically asymmetric and its centre of mass is different from the point of support, then the equations of rotation do not have a third analytic integral in the form of series (1.4) and which is independent of the integrals $H=T+\varepsilon V$ and $F$. As already mentioned in § 2, the Hamiltonian of the heavy top rotation problem in the action–angle variables of the Euler–Poinsot problem has the form (2.1). This fact was well known to Poincaré (see § 86 in [1]), although he did not written the functions $H_0$ and $H_1$ explicitly (in the actual fact, this was not necessary for his analysis). In place of explicit formulas for the Hamiltonian of the unperturbed problem $H_0$, Poincaré used the geometric Poinsot representation. The form of the expansion of the perturbing function $H_1$ as a Fourier series was related by Poincaré to the Jacobi formulas (which he knew of), which represent the direction cosines of a moving frame in the Euler–Poinsot problem in terms of elliptic functions. Explicit formulas (as well as the necessary references) for the Hamiltonian of the perturbed Euler–Poinsot problem can be found, for example, in Chap. 2 of [5]. The function $H_0$ is a homogeneous function of $y_1$ and $y_2$ of degree $2$. It should be pointed out that $y_2 \geqslant 0$ is the kinetic moment of the top. Hence $|y_3| \leqslant y_2$. However, we also have the inequality $|y_1| \leqslant y_2$. Let $A$, $B$, $C$ be the principal moments of inertia of the rigid body. In the asymmetric case, it can be assumed that $A<B<C$. The function $H_0(y_1,y_2)$ is analytic in the domain
$$
\begin{equation*}
\Delta_a=\Delta \setminus (\{y_1=0\} \cup \{2H_0=B^{-1}y_2^2\}\cup \{|y_1|=y_2\}),
\end{equation*}
\notag
$$
where $\Delta=\{y_1,y_2\colon y_2 \geqslant 0,\, |y_1| \leqslant y_2\}$. According to [3], the Euler–Poinsot problem is non-degenerate, that is, the Hessian of the function $H_0$ is positive in the two connected subdomain of $\Delta_a$ adjacent to the straight lines $|y_1|=y_2$; the Hessian is negative in the remaining two subdomains. In addition, $dH_0 \ne 0$ in the domain $\Delta_a$. So, conditions 1) and 3) of Theorem 2 are met. Let us now discuss the fourth condition of Theorem 2. By the Jacobi formulas (see Chap. 2 in [5]),
$$
\begin{equation}
h_0=\frac{y_3}{y_2}\,\frac{\pi}{2K}\, \frac{\varkappa}{\sqrt{\varkappa^2+\Lambda^2}}\,\frac{z}{r},
\end{equation}
\tag{3.1}
$$
where
$$
\begin{equation*}
\varkappa^2=\frac{C(A-B)}{A(B-C)},\qquad \Lambda^2=\frac{2CH_0-y_2^2}{y_2^2-2AH_0},
\end{equation*}
\notag
$$
$r=(x^2+y^2+z^2)^{1/2}$, $K$ is the complete elliptic integral of the first kind of modulus $\Lambda<1$, and $x$, $y$, $z$ are the coordinates of the centre of mass of the body relative to the principal moments of inertia of the rigid body. In the domain $\Delta_a$, if $z \ne 0$. It is worth noting that the action–angle variables in the Euler–Poinsot problem can be introduced in various ways. The inertia axis of the top with momentum of inertia $C$ was singled out for the introduction of the variables employed for derivation of (3.1). Formula (3.1) will slightly change if this axis is replaced by a different inertia axis; in particular, the $x$- or $y$-coordinate will appear in place of the $z$-coordinate of the centre of mass. So, condition 4) is also met. Let us now discuss the second condition in Theorem 2. The Hamiltonian function of the problem on a heavy asymmetric top is defined and analytic in the variables $y_1$ and $y_2$ in the domain So, the domain $D$ depends on the constant in the area integral $F$. In particular, all the coefficients of the Fourier function $H_1$ are analytic in the domain $D$.Let us find the structure of the secular set. It turns out that, in each of the four connected subdomains of the domain $D$, the set $\mathscr{B}$ consists of a countable number of straight lines passing through the origin. These straight lines are accumulated near one of the two straight lines given by the equation these straight lines lie on the boundary of the domain $D$. For a detailed account of this situation,1 see [5], Chap. 3, § 1.Unfortunately, condition 2) of Theorem 2 is not met, even though the number of straight lines in the secular set $\mathscr{B}$ is infinite. If the set (3.3) (comprising, in particular, unstable constant rotations about the middle inertia axis) were inside the domain of analyticity of the Hamiltonian $H_0+\varepsilon H_1$, then $\mathscr{B} \cap D$ would be a key set for the class of analytic functions $\mathscr{A}(D)$. However this is not so. This difficulty, which is also present in the problem of an additional integral of the reduced system, was overcome in [4] (see also [5], Chap. 3, § 3, § 4). The idea here is to consider the secular set $\mathscr{B}$ as a set of resonant tori of the unperturbed problem in the phase space; these tori, which correspond to the action variables $(y_1,y_2) \in \mathscr{B}$, are accumulated near the separatrices of unstable rotations about the middle inertia axis of the top. The set of these tori form the key set for the class of functions analytic in the entire phase space. A combination of this idea with the method of the proof of Theorem 2 establishes Theorem 3. A new essential point here is inequality (3.2), the remaining steps proceed as in the proof of non-existence of an additional analytic integral of the reduced system (see Chap. 3 in [5]). § 4. Momentum-polynomial integralsThe energy integral $H$ and the area integral $F$ are momentum polynomials of degree $2$ and $1$, respectively. All the coefficients of these polynomials are single-valued analytic functions on the configuration manifold $G=\operatorname{SO}(3)$. Theorem 4. If the centre of mass of a heavy asymmetric top is different from the point of support, then the equations of rotation of this top do not admit a third integral in the form of a momentum-polynomial which is single-valued and analytic on the phase space and independent of the integrals $H$ and $F$. This result is not reduced to Husson’s theorem on non-existence of an additional algebraic integral of the Euler–Poisson equations for the following reasons: 1) the additional integral in Theorem 4 is a polynomial in momenta, rather than in all variables; 2) the integrals $F$ and $\Phi$ are not assumed to be involutive. In the actual fact, Theorem 3 and 4 are equivalent. Indeed, suppose that a Hamiltonian system with Hamiltonian function $H=T(y,x)+\varepsilon V(x)$ has an integral in the form of a series in powers of the small parameter $\varepsilon$. The change of variables
$$
\begin{equation}
x \mapsto x,\qquad y \mapsto \sqrt{\varepsilon}\,y,\qquad t \mapsto \frac{t}{\sqrt{\varepsilon}}
\end{equation}
\tag{4.2}
$$
transforms the equations with Hamiltonian $T+\varepsilon V$ to equations with Hamiltonian $T+V$, and integral (4.1) will change to the function
$$
\begin{equation*}
\sum\Phi_k(\sqrt{\varepsilon}\,y,x)\varepsilon^k= \sum \widehat{\Phi}_m(y,x)(\sqrt{\varepsilon}\,)^m,
\end{equation*}
\notag
$$
where $\widehat{\Phi}_m$ are some polynomials in momenta $y$. The new Hamiltonian system is independent of $\varepsilon$, and hence the polynomials $\widehat{\Phi}_m$ are integrals of this system. Of course, not all these polynomials are independent. However, if the functions $H$ and $\Phi$ (qua series in powers of $\varepsilon$) are independent, then among the polynomials $\{\widehat{\Phi}_m\}$ there is at least one independent of the function $T+V$. A similar remark also applies to the three integrals $H$, $F$, and $\Phi$. The converse assertion also holds: if a Hamiltonian system with Hamiltonian $T+V$ admits a momentum-polynomial integral, then the Hamiltonian system with Hamiltonian $T+\varepsilon V$ has a power-series integral (4.1). The proof is by the inverse change of variables to (4.2). A similar approach was used in classical papers on algebraic integrals of quasihomogeneous systems of differential equations (see, for example, [2]). Theorems 3 and 4 suggest the following natural question: Is it true that, if the centre of mass is different from the point of support, then, in the dynamically symmetric case, the equations of rotation admit a third integral analytic in the six-dimensional phase space only in the classical cases of Lagrange and Kovalevskaya? Here, it is not necessary at all that the third integral should be in involution with the area integral. For the reduced system with two degrees of freedom, the problem of existence of additional integrals was solved in [6] (this paper considers the integrals which are meromorphic in the complexified phase space), and also partially in [7]–[11] (which deal with real analytic integrals). A complete solution of the problem on real analytic integrals for the reduced system in the heavy top problem was given in [12]. The proof depends on the method of [6]; the dynamical effects that are obstructions for complete integrability are indicated in [7]–[11]. A numerical analysis of the chaotic behaviour of the reduced system in non-integrable cases can be found in [13]. Non-existence of new algebraic integrals in the dynamically symmetric case was proved in [14] (for simplifications and correction of inaccuracies, see [15]). 
Citation in format AMSBIB
\Bibitem{Koz24} |
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