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This article is cited in 6 scientific papers (total in 6 papers) On the integrability of the equations of dynamics in a non-potential force field V. V. Kozlov Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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     To I. A. Taimanov on his 60th birthday 1. Circulatory systemsLet $M^n=\{x_1,\dots,x_n\}$ be the configuration space of a mechanical system with $n$ degrees of freedom, where $x=(x_1,\dots,x_n)$ are the generalized coordinates. Let be the kinetic energy and $F=(F_1,\dots,F_n)$ be the generalized forces. The dynamics of such a system is described by Lagrange’s differential equations
$$
\begin{equation}
\frac{d}{dt}\,\frac{\partial T}{\partial\dot{x}_j}- \frac{\partial T}{\partial x_j}=F_j,\qquad 1 \leqslant j \leqslant n.
\end{equation}
\tag{1.1}
$$
If the force $F$ depends only on the position of the system, then the system is called positional. If, moreover, the differential 1-form is exact (that is, it is the total differential of some function $U$ on $M^n$), then such a force is called potential, and the mechanical system is conservative. In this case differential equations (1.1) admit the energy integral $T-U$, which is quadratic in the velocity.A mechanical system that is under the action of positional but not potential forces is often called circulatory. In between conservative and circulatory systems are ‘quasi-conservative’ systems, where the work of the forces (1.2) is a closed but not exact 1-form. In this case the force function $U$ is a ‘multivalued’ function on the configuration space, and the energy integral $T-U$ is a ‘multivalued’ function on the phase space. A generalization of Helmholtz’s theorem on the decomposition of a positional force into a circulatory force and a potential force was discussed in [1]. In [2] and [3] a classification of generalized forces with respect to their proximity to potential forces was given. In general, according to Hodge’s theory [4], the 1-form (1.2) can be represented as a sum of three forms, where $\alpha$ is a function on the configuration space, $\beta$ is a 2-form, and $\gamma$ is a harmonic 1-form. Unlike the external differential $d$, the operation of divergence $\delta$ does not increase, but decreases the degree of the form by one. The harmonic form $\gamma$ satisfies the condition $(d\delta+\delta d)\gamma=0$. In the case of a closed configuration space the decomposition (1.3) is unique. The divergence operation $\delta$ (like the decomposition (1.3) itself) depends on the choice of the Riemannian metric on $M^n$. In problems of mechanics it is natural to choose as the Riemannian metric the ‘intrinsic’ metric given by the kinetic energy of the mechanical system. This range of questions was discussed in [5] from the point of view of the problem of the stability of equilibrium states.The form $d\alpha$ in (1.3) corresponds to a potential force, and $\gamma$ represents a quasi- potential force field (of course, provided that the first cohomology group of configuration space $M^n$ is non-trivial; otherwise $\gamma=0$). If the 1-form of work of the forces (1.2) is represented by the sum $d\alpha+\gamma$, then the force $F$ is also quasi-potential. Finally, the term $\delta\beta$ in (1.3) corresponds to the circulatory force. As a simple example, consider the case of a flat torus: $M=\mathbb{T}^n=\{x_1,\dots,x_n \, \operatorname{mod} 2\pi\}$, and the metric is given by the kinetic energy $T=(\dot{x},\dot{x})/2$. Let us expand the positional force in a Fourier series: The coefficients are vectors from $\mathbb{C}^n$, with $F_{-k}=\overline{F}_{k}$. When $k \ne 0$, these vectors are uniquely represented as the sum of $F'_k$ and $F_k^{\prime\prime}$, where $F'_k$ is collinear to the vector $k$, and $F_k^{\prime\prime}$ is orthogonal to $k$. We have where
$$
\begin{equation*}
F'=\sum_{k \ne 0}F'_k e^{i(k,x)}\quad\text{and}\quad F^{\prime\prime}=\sum_{k \ne 0}F^{\prime\prime}_k e^{i(k,x)}.
\end{equation*}
\notag
$$
The decomposition (1.4) corresponds to the Hodge decomposition (1.3). The force $F^{\prime\prime}$ is circulatory: its divergence is zero. Indeed,
$$
\begin{equation*}
\sum_{j=1}^n \frac{\partial F^{\prime\prime}_j}{\partial x_j}= \sum_{k \ne 0}i(F^{\prime\prime}_k,k) e^{i(k,x)}=0.
\end{equation*}
\notag
$$
On the contrary, the force $F'$ is potential. Since $F'_k=ku_k$, $u_k \in \mathbb{C}$ ($\overline{u}_k=u_{-k}$), the force function $U$ is The constant force $F_0$ is quasi-potential: if $F_0 \ne 0$, then the corresponding force function $(F_0,x)$ is a multivalued harmonic function on $\mathbb{T}^n$. In the theory of circulatory systems one usually considers problems of stability and bifurcations of equilibrium states and stationary motions, taking the additional gyroscopic and dissipative forces into account (see, for instance, [6], [7] and the references there). The first steps in the problem of the exact integration of differential equations describing the dynamics of circulatory systems were made in [8] and [9]. More precisely, the conditions for the existence of polynomial in velocity first integrals with coefficients which are single-valued on $M^n$ have been investigated. The question of their existence depends on the topology of the configuration space. Theorem 1 (see [9]). If the genus of the surface $M^2$ is greater than one, then the equations of motion admit a single-valued non-constant polynomial first integral if and only if the force field is potential. The proof is based on a result on the non-existence of an additional polynomial integral of geodesic flows on a closed surface whose genus is greater than one [10], [11]. A multidimensional version of the theorem on topological obstacles to the complete integrability of geodesic flows was obtained in [12] and [13]. It would be interesting to relate this result to the non-existence of non-trivial single-valued polynomial integrals of multidimensional circulatory systems. Let $n=2$ again, and let the configuration space be a two-dimensional torus $\mathbb{T}^2=\{x_1,x_2\, \operatorname{mod}{2\pi}\}$. Assume that the kinetic energy is reduced to the conformal form where $\Lambda\colon \mathbb{T}^2 \to \mathbb{R}$ is a smooth positive function. Let us expand the conformal factor $\Lambda$ in a double Fourier series
$$
\begin{equation*}
\sum \lambda_k e^{i(k,x)},\qquad k=(k_1,k_2) \in \mathbb{Z}^2.
\end{equation*}
\notag
$$
If
Theorem 2 (see [9]). Assume that the equations of motion admit a non-constant integral with smooth coefficients on $\mathbb{T}^2$ which is polynomial in velocities. If conditions (A) and (B) hold, then the force field $F$ is potential. Equations (1.1) can be represented in Hamiltonian variables on the cotangent bundle $T^*M$:
$$
\begin{equation}
\dot{x}=\frac{\partial H}{\partial y}\,,\qquad \dot{y}=-\frac{\partial H}{\partial x}+F(x),
\end{equation}
\tag{1.5}
$$
where
$$
\begin{equation*}
y=(y_1,\dots,y_n),\qquad y_j=\frac{\partial T}{\partial \dot{x}_j}\,,
\end{equation*}
\notag
$$
is the ‘canonical’ momentum of the mechanical system at the point $x \in M^n$ (an element of the vector space $T_x^*M^n$). The function $H$ is the kinetic energy represented in the variables $x$ and $y$. Equations (1.5) are, of course, not Hamiltonian in general. However, under some special conditions, they can be brought into the Hamiltonian form even when the force $F$ is not potential [8].
2. Euler–Jacobi–Lie theoremAccording to the general approach, to integrate an autonomous system of differential equations in the $m$-dimensional phase space $\Gamma^m=\{z\}$ we need to know $m$ independent tensor invariants which are in certain natural relations to one another (see the discussion in [16]). In particular, the vector field $v$ defining this dynamical system is one of the invariants.The equations of motion (1.5) of a circulatory system admit another tensor invariant, the differential $2n$-form of phase volume This observation goes back to Jacobi [17]. Thus, for the complete integrability of equations (1.5), $2n-2$ other independent tensor invariants are needed. For example, these missing invariants can be $2n-2$ functionally independent first integrals (scalar invariants). Then the equations of motion are exactly integrable as guaranteed by the classical Euler–Jacobi theorem about the last multiplier [17]. In particular, in the case of two degrees of freedom (when $n=2$) it is sufficient to know two independent first integrals.A more general approach is based on the application of the Euler–Jacobi–Lie theorem (EJL theorem in what follows) [18]. Let us recall its formulation. Assume that the system of differential equations (2.1) admits an invariant volume $m$-form and also $k$ integrals $f_1,\dots,f_k$ and $l$ commuting symmetry vector fields (all commutators $[v,u_s]$ and $[u_s,u_r]$ are zero). It is assumed that the integrals $\{f_j\}$ are functionally independent, and the vectors $v,u_1,\dots,u_l$ are linearly independent at all points in the region under consideration in the phase space. In addition, assume that
$$
\begin{equation}
L_{u_j}f_s=\biggl(\frac{\partial f_s}{\partial z}\,,u_j\biggr)=0
\end{equation}
\tag{2.3}
$$
for all $1\leqslant j \leqslant l$ and $1\leqslant s \leqslant k$, and also for all $1\leqslant j \leqslant l$ ($L_u$ denotes the Lie derivative with respect to the vector field $u$). If $k+l=m-2$, then the differential equations (2.1) are integrable by quadratures. In [18] this statement was proved under more general assumptions about the commutators of the symmetry fields $\{u_j\}$. Conditions (2.3) and (2.4) mean that the phase flows generated by the fields $\{u_j\}$ preserve both the functions $\{f_s\}$ and the volume form $\Omega$. If $l=0$, then we obtain the Euler–Jacobi theorem. Conditions (2.4) can be represented in an equivalent form:
$$
\begin{equation*}
\operatorname{div}(\rho u_j)=0,\qquad 1\leqslant j \leqslant l.
\end{equation*}
\notag
$$
Returning to systems in a non-potential force field, let us discuss the simplest, but important case of two degrees of freedom. According to the general EJL theorem, exact integration is possible in the following three cases:
The qualitative properties of phase flows in these three cases of integrability are essentially different. Let us emphasize that for a system in a non-potential force field the presence of a symmetry field does not imply the existence of a first integral linear in velocities (as in Noether’s classical theorem). For example, let the kinetic energy and force field components be independent of $x_1$. Then the vector field given by the differentiation operator $\partial/\partial x_1$ is obviously a symmetry field. However, the corresponding momentum $y_1$ is conserved only under the additional condition $F_1=0$. 3. Example of the Euler–Jacobi integrabilityConsider a system with kinetic energy in the form of a Liouville metric
$$
\begin{equation}
T=\frac{1}{2}\,\frac{A_1 y_1^2+A_2 y_2^2}{B_1-B_2}\,,
\end{equation}
\tag{3.1}
$$
where the functions $A_k$ and $B_k$ only depend on one coordinate $x_k$ ($k=1,2$). As shown in [8], if the components of the positional force have the form
$$
\begin{equation}
F_1=-\frac{1}{B_2}\,\frac{\partial f}{\partial x_1}\quad\text{and}\quad F_2=-\frac{1}{B_1}\,\frac{\partial f}{\partial x_2},\qquad
\end{equation}
\tag{3.2}
$$
then the equations of motion admit the quadratic integral where
$$
\begin{equation}
\Phi_2=\frac{1}{2}\,\frac{B_1A_2y_2^2+B_2A_1y_1^2}{B_1-B_2}\,.
\end{equation}
\tag{3.3}
$$
In the general case the force (3.2) is, of course, not potential. In [8] the question of when a mechanical system with kinetic energy (3.1) in the force field (3.2) is conformally Hamiltonian was considered. Thus we can specify cases where the equations of motion can be integrated by separating the variables. First, following [8] we present the equations of motion in the following quasi- Hamiltonian form:
$$
\begin{equation}
B_2 \dot{y}_1=-\frac{\partial\mathcal{K}}{\partial x_1}\,,\quad B_2 \dot{x}_1=\frac{\partial\mathcal{K}}{\partial y_1}\,;\qquad B_1 \dot{y}_2=-\frac{\partial\mathcal{K}}{\partial x_2}\,,\quad B_1 \dot{x}_2=-\frac{\partial\mathcal{K}}{\partial y_2}\,.
\end{equation}
\tag{3.4}
$$
We perform a time change $t \mapsto \tau$ using the formula and make the change of coordinates $x_1 \mapsto q_1$, $x_2 \mapsto q_2$ such that
$$
\begin{equation}
q_1=\int_{x_1^0}^{x_1}B_1^{-1}(s)\,ds\quad\text{and}\quad q_2=\int_{x_2^0}^{x_2}B_2^{-1}(s)\,ds.
\end{equation}
\tag{3.5}
$$
Let us introduce another ‘Hamiltonian’
$$
\begin{equation}
\mathcal{H}(q_1,y_1,q_2,y_2)=\mathcal{K}(x_1,y_1,x_2,y_2),
\end{equation}
\tag{3.6}
$$
where $x_1$ and $x_2$ are expressed in terms of $q_1$ and $q_2$ by the inversion of formulae (3.5). After these substitutions, equations (3.4) take the canonical form of Hamilton’s equations:
$$
\begin{equation}
\frac{dy_k}{d\tau}=-\frac{\partial\mathcal{H}}{\partial q_k}\,,\quad \frac{dq_k}{d\tau}=\frac{\partial\mathcal{H}}{\partial y_k}\,,\qquad k=1,2.
\end{equation}
\tag{3.7}
$$
Using (3.3) the new Hamiltonian $\mathcal{H}$ is reduced to a form suitable for the separation of variables:
$$
\begin{equation}
\mathcal{H}=\frac{1}{2}\, \frac{A_2B_2^{-1}y_2^2+A_1B_1^{-1}y_1^2}{B_2^{-1}-B_1^{-1}}+f.
\end{equation}
\tag{3.8}
$$
Here the variables $x_1$ and $x_2$ must be replaced by $q_1$ and $q_2$ by formulae (3.5). Clearly, the products $A_kB_k^{-1}$ ($k=1,2$) (as well as the functions $B_k^{-1}$) depend only on the $q_k$. In particular, if in the expressions for components of the external force (3.2) the function $f$ is of the form
$$
\begin{equation}
\frac{C_1(x_1)+C_2(x_2)}{B_2^{-1}(x_2)-B_1^{-1}(x_1)}\,,
\end{equation}
\tag{3.9}
$$
then the canonical equations (3.7) with Hamiltonian (3.8) are solved by separating the pairs of variables $q_1$, $y_1$ and $q_2$, $y_2$. Thus, we have an integrable (in the sense of Euler–Jacobi) mechanical system with kinetic energy (3.1) and non-potential positional force (3.2), (3.9). In this case the equations of motion admit two independent integrals, which are quadratic in momentum. An explicit integration of the Hamiltonian system (3.7)–(3.9) with separated variables can be performed using known methods (see, for instance, [19] and [20]). First, following [19] the angular variables $\psi_1$, $\psi_2\, \operatorname{mod} 2\pi$ are introduced on two- dimensional invariant tori (so that the coordinate $q_k$ becomes a function of $\psi_k$ alone) in which the equations of motion (3.7) take the following form:
$$
\begin{equation}
\frac{d\psi_k}{d\tau}=\frac{\omega_k}{b_2^{-1}-b_1^{-1}}\,,\qquad k=1,2.
\end{equation}
\tag{3.10}
$$
Here $\omega_1,\omega_2=\operatorname{const}$ (they depend on constant quadratic integrals), and the $b_k$ are the functions $B_k(q_k)$ in which $q_k$ is expressed in terms of the variables $\psi_k\, \operatorname{mod} 2\pi$. The variables $\psi_1$ and $\psi_2$ in (3.10) are separated: the denominator is the sum of two functions depending only on $\psi_1$ and $ \psi_2$, respectively. Hence [13] there is a change of angular variables
$$
\begin{equation*}
\psi_1,\psi_2\, \operatorname{mod} 2\pi \mapsto \varphi_1,\varphi_2\, \operatorname{mod} 2\pi
\end{equation*}
\notag
$$
that transforms system (3.10) into
$$
\begin{equation*}
\frac{d\varphi_k}{d\tau}=\frac{\omega_k}{\lambda}\,,\qquad \lambda=\frac{1}{2\pi}\int_0^{2\pi}b_2^{-1}(s)\,ds- \frac{1}{2\pi}\int_0^{2\pi}b_1^{-1}(s)\,ds.
\end{equation*}
\notag
$$
So the Hamiltonian system (3.7) is linearized on the invariant tori: the new angular variables $\varphi_1$ and $\varphi_2$ are linear functions of the new time $\tau$. On the other hand this fact also follows from the general geometric Liouville theorem about completely integrable Hamiltonian systems. It is less obvious that the flow of the initial mechanical system in the non- potential force field (with kinetic energy (3.1) and external field (3.2), (3.9)) is also linearized on the same invariant tori with respect to the ‘old’ time $t$. Indeed, using the formula $dt=b_1b_2\,d\tau$ the system of differential equations is transformed into the following form:
$$
\begin{equation}
\dot{\psi}_k=\frac{\omega_k}{b_1(\psi_1)-b_2(\psi_2)}\,,\qquad k=1,2.
\end{equation}
\tag{3.11}
$$
Here the variables $\psi_1$ and $\psi_2$ are also separated. Therefore, in some new angular coordinates $\nu_1$, $\nu_2\, \operatorname{mod} 2\pi$ the system of equations (3.11) is reduced to
$$
\begin{equation}
\dot{\nu}_k=\frac{\omega_k}{\mu}\,,\qquad \mu=\frac{1}{2\pi}\int_0^{2\pi}b_1(s)\,ds- \frac{1}{2\pi}\int_0^{2\pi}b_2(s)\,ds.
\end{equation}
\tag{3.12}
$$
The systems of differential equations (3.10) and (3.11) are dual: the time substitution $d\tau'=b_1^{-1}b_2^{-1}\,dt$ transforms (3.11) into (3.10) (since $d\tau'=d\tau$). Such time substitutions belong to the Liouville class [21]: they preserve the property of uniform motion along windings on the invariant tori. Liouvillean time substitutions in algebraically integrable Hamiltonian systems were investigated in [21]. Since the equations of motion of a mechanical system with kinetic energy (3.1) and external forces (3.2), (3.9) are reduced on the invariant tori to equations (3.12) (with constant right-hand sides), they turn out (as shown in [22]) to be Hamiltonian in neighbourhoods of these tori. Thus, the equations of motion with kinetic energy (3.1) and forces (3.2) reduce to the conformal Hamiltonian form [8]. And if the function $f$ in formulae (3.2) for the force field components is of the form (3.9), then these equations are Hamiltonian (at least on the open union of invariant tori). 4. An invariant torus without quasiperiodic motionsLet us give a simple example of a circulatory system with toric configuration space $M=\mathbb{T}^n=\{x_1,\dots,x_n\, \operatorname{mod} 2\pi\}$ which has an invariant torus with everywhere dense trajectories, but the flow on the torus is not reduced to a quasiperiodic motion. Let $T=\frac12\sum y_k^2$, and assume that the generalized forces have the form
$$
\begin{equation}
F_k(x)=\omega_k f\sum_{i=1}^n \frac{\partial f}{\partial x_i}\omega_i,\qquad 1 \leqslant k \leqslant n.
\end{equation}
\tag{4.1}
$$
Here $\omega_1,\dots,\omega_n$ is a non-resonant set of real numbers (if $\sum k_j \omega_j=0$ for some integers $k_j$, then $k_1=\cdots=k_n=0$), and $f$ is a smooth positive function on $\mathbb{T}^n$. Assume that
$$
\begin{equation*}
\Phi_k=y_k-\omega_k f,\qquad 1 \leqslant k \leqslant n.
\end{equation*}
\notag
$$
Then the time derivatives of these functions with respect to the corresponding system of equations (1.5) are
$$
\begin{equation*}
\dot\Phi_k=-\Phi_k\sum_{i=1}^n \frac{\partial f}{\partial x_i}\,\omega_i,\qquad 1 \leqslant k \leqslant n.
\end{equation*}
\notag
$$
Hence the $n$-dimensional surface is a smooth invariant manifold. It is projected one-to-one onto the configuration torus $\mathbb{T}^n=\{x\}$. Therefore, the manifold (4.2) itself is an $n$-dimensional torus parameterized by the angular coordinates $x_1,\dots,x_n$. The equations of motion in these coordinates have the following form:
$$
\begin{equation}
\dot x_k=\omega_k f(x),\qquad 1 \leqslant k \leqslant n.
\end{equation}
\tag{4.3}
$$
Such equations were considered in [23]–[25] from various points of view. As is well known [24], [26], for almost all frequencies $\omega_1,\dots,\omega_n$, using a suitable invertible change of angular variables $x\, \operatorname{mod} 2\pi \mapsto z\, \operatorname{mod} 2\pi$ equations (4.3) can be reduced to
$$
\begin{equation*}
\dot z_k=\frac{\omega_k}{\lambda}\,,\qquad \lambda=\frac{1}{(2\pi)^n}\int_{\mathbb{T}^n}\frac{d^nx}{f(x)}\,.
\end{equation*}
\notag
$$
So on invariant torus we have a quasiperiodic motion. On the other hand, for a suitable choice of the frequencies $\{\omega\}$ and the function $f$ the phase flow of the system of differential equations (4.3) cannot be reduced to a quasiperiodic motion (see, for instance, [24] and [27]). The general properties of the dynamical system (4.3) on an $n$-dimensional torus were investigated in [27] (where other references on this topic can also be found). Let us show that, given the assumptions made above about the set $\{\omega\}$ and the non-constant function $f$, the force with components (4.1) is not potential. Indeed,
$$
\begin{equation*}
F_k=\omega_k\mu\quad\text{and}\quad \mu=\frac{1}{2}\sum\frac{\partial f^2}{\partial x_i}\,\omega_i.
\end{equation*}
\notag
$$
The condition for this force to be potential is as follows:
$$
\begin{equation*}
\frac{\partial F_k}{\partial x_l}=\frac{\partial F_l}{\partial x_k} \quad\Longleftrightarrow\quad \omega_k\frac{\partial \mu}{\partial x_l}= \omega_l\frac{\partial \mu}{\partial x_k}\,.
\end{equation*}
\notag
$$
Since the set of frequencies $\omega_1,\dots,\omega_n$ is non-resonant, the ratio $\omega_k/\omega_l$ is irrational for all $k \ne l$. Hence the function $\mu$ is independent of any pair of angular coordinates. Thus, $\mu=\operatorname{const}$. By averaging both parts of the equality
$$
\begin{equation*}
\mu=\frac{1}{2}\sum\frac{\partial f^2}{\partial x_i}\omega_i
\end{equation*}
\notag
$$
over $\mathbb{T}^n$ we conclude that $\mu=0$. But then $f^2$ is a first integral of system (4.3). Since all of its trajectories are everywhere dense on $\mathbb{T}^n$, it follows that $f=\operatorname{const}$. However, this contradicts the original assumption.
5. Non-Noether symmetriesFirst we give a simple example of a circulatory system that is integrable according to case (B) in § 2. Let the configuration space be the 2-dimensional torus $\mathbb{T}^2=\{x_1,x_2\, \operatorname{mod} 2\pi\}$, and let the kinetic energy be ‘Euclidean’: $T=(\dot x_1^2+\dot x_2^2)/2$. The equations of motion are Newton’s equations Assume that the components $F_1$ and $F_2$ of the force field depend on the coordinate $x_1$ only. If $F_2\ne\operatorname{const}$, then this force is non-potential.If the mean of $F_1$ on the circle $x_1\, \operatorname{mod} 2\pi$ is zero, then we can set where $a$ is a $2\pi$-periodic function of $x_1$. Hence equations (5.1) admit a single-valued first integral If the mean of $F_1$ is non-zero, then there is also a quadratic integral, but it is a multivalued function on the phase space.In addition, there is an obvious symmetry field $u$ with differentiation operator $L_u=\partial/\partial x_2$. Since the right-hand side of (5.1) does not depend on the $x_2$-coordinate, it is obvious that
$$
\begin{equation*}
[u,v]=0,\quad L_u\Phi=0,\quad\text{and}\quad L_u\Omega=0.
\end{equation*}
\notag
$$
Thus, all conditions of the EJL theorem on integrability are satisfied. The explicit integration of (5.1) is quite elementary. If $F_2 \ne 0$, then the vector field $u$ is not of Noether type: it does not generate a first integral that is linear in the momentum. Let us now discuss the structure of the phase flow of a system that is integrable in accordance with the assumptions of case (B). Let $c$ be a non-critical value of the given integral $\Phi\colon \Gamma\to \mathbb{R}$, and let $I_c=\{x,y\colon\Phi(x,y)=c\}$ be the corresponding 3-dimensional invariant manifold, which we assume to be connected. Since the phase flow of the system preserves the non-degenerate 4-form $\Omega$ and $\Phi$ is a first integral, it is well known that the restriction of the flow to $I_c$ preserves a certain volume 3-form $\omega$ on $I_c$. Next, consider the differential 1-form Since the vectors $u$ and $v$ are assumed to be linearly independent at each point of $\Gamma$ (hence also on $I_c$), $\psi$ is a non-trivial 1-form. As the vector fields $u$ and $v$ commute, this form is closed:
$$
\begin{equation*}
\begin{aligned} \, d\psi&=di_u(i_v \omega)=L_u i_v \omega-i_u(di_v \omega) \\ &=i_v L_u \omega-i_u(L_v \omega-i_v\,d\omega)=0 \end{aligned}
\end{equation*}
\notag
$$
($L_u \omega=L_v \omega=0$ since the form $\omega$ is invariant, and $d\omega=0$ since it is closed). Thus, $\psi$ represents locally the differential of a smooth function on $I_c$. Furthermore, if (in particular, if $I_c$ is simply connected), then $\psi=df$, where $f$ is a smooth function on the whole of $I_c$. By (5.2) $f$ is a first integral for the vector fields $v$ and $u$. Let the level surfaces of the function $f\colon I_c \to \mathbb{R}$ be compact (this holds if the 3-dimensional manifold $I_c$ is compact). Then regular level surfaces of $f$ are two-dimensional tori with independent and commuting tangent vector fields $v$ and $u$. Consequently, the flows of the circulatory system on these tori reduce to quasiperiodic motions. If condition (5.3) is not fulfilled, then $f$ is ‘multivalued’ on $I_c$: its level surfaces are not closed. Under the condition of regularity ($df \ne 0$) they are homeomorphic to two-dimensional cylinders or planes. In either case the phase flows on three-dimensional invariant manifolds $I_c$ do not exhibit chaotic behavior. In conclusion, a few words about the behaviour of integrable systems in case (C) from § 2 are in order. Consider the non-zero 1-form
$$
\begin{equation}
\Psi=i_v i_{u_1} i_{u_2}\Omega=\Omega(v,u_1,u_2,\,\cdot\,),
\end{equation}
\tag{5.4}
$$
where $u_1$ and $u_2$ are commuting symmetry fields. We assume that the vectors $v$, $u_1$, and $u_2$ are linearly independent. Then $d\Psi=0$, and so $\Psi=df$ locally, where $f$ is a first integral of the equations of motion. If $H^1(\Gamma,\mathbb{R})=0$, then the smooth function $f$ is globally defined on the phase space $\Gamma$. Let the 1-form $\Psi$ be exact. Then the connected compact invariant surfaces $\{x,y\colon f(x,y)=c\}$ are three-dimensional tori with quasiperiodic motion. The general case should be considered in terms of the theory of foliations. As an illustrative example, consider equations (5.1) in the phase space $\Gamma=\mathbb{T}^2\times \mathbb{R}^2$, and assume that the force components $F_1$ and $F_2$ are constant. The commuting non-Noether symmetry fields $u_1$ and $u_2$ are defined by the differentiation operators $\partial/\partial x_1$ and $\partial/\partial x_2$. According to (5.4), the 1-form $\Psi$ is the differential of the function $f=F_2y_1-F_1y_2$. If $F_1^2+F_2^2 \ne 0$, then $df \ne 0$. However, three-dimensional invariant surfaces $\{f=c\}$ are not compact: they are diffeomorphic to the cylinders $\mathbb{T}^2\times\mathbb{R}$. There are three commuting tangent vector fields, $v$, $u_1$, and $u_2$ on these manifolds. Consequently, it is possible to choose two angular coordinates and one linear coordinate so that they vary uniformly over time. For example, when $F_2=0$, the equations of motion on the manifold $\{f=c\}$ have the form
$$
\begin{equation*}
\dot x_1=y_1,\quad \dot y_1=F_1,\quad \dot x_2=-\frac{c}{F_1}\,.
\end{equation*}
\notag
$$
Introducing the new angular coordinate $\widetilde{x}_1=x_1-y_1^2/(2F_1)$ we obtain the missing equation $(\widetilde{x}_1)^{\boldsymbol{\cdot}}=0$.
6. Systems with a third-degree integralA review of known results on first integrals of third degree in momenta of conservative systems with two degrees of freedom can be found, for example, in [28], Chap I, § 3. Among them there are no systems with smooth periodic potential energy, and if the potential is a function on a two-dimensional torus, then it necessarily has singularities in the form of poles. It turns out that this is not a coincidence: if a system with toric configuration space, Euclidean kinetic energy, and smooth potential admits an integral of degree three in the momentum, then there must also be an integral of the first degree. This result was established in [29]; for generalizations, see [30]. For systems in a non-potential force field this is not true. We demonstrate this using the example of equations (5.1) with angular coordinates $x_1$, $x_2\, \operatorname{mod}{2\pi}$. Assume that
$$
\begin{equation}
F_1=\frac{\partial a}{\partial x_1}\quad\text{and}\quad F_2=-a\frac{\partial a}{\partial x_1}\,,
\end{equation}
\tag{6.1}
$$
where $a$ is a smooth non-constant function of $x_1$. Equations (5.1) admit an obvious quadratic integral and a third-degree integral The functions $f_1$ and $f_2$ are independent: the rank of their Jacobian matrix is equal to 2 almost everywhere. We show that equations (5.1) with force (6.1) do not admit other integrals which are quadratic in the momentum. In particular, there are no linear first integrals. As shown in [8], a quadratic integral has the form
$$
\begin{equation}
\frac{1}{2}(\alpha y_1^2+2\beta y_1y_2+\gamma y_2^2)+g,
\end{equation}
\tag{6.2}
$$
where $g\colon \mathbb{T}^2\to\mathbb{R}$ is smooth function. The homogeneous part of (6.2) is an integral in the problem of the inertial motion along the torus. It is easy to show that then the coefficients $\alpha$, $\beta$, and $\gamma$ are constants. Further, according to [8], the following equation holds:
$$
\begin{equation*}
\begin{bmatrix} \alpha & \beta \\ \beta & \gamma \end{bmatrix} \begin{bmatrix} F_1 \\ F_2 \end{bmatrix}=\begin{bmatrix} \dfrac{\partial g}{\partial x_1} \\ \dfrac{\partial g}{\partial x_2} \end{bmatrix}.
\end{equation*}
\notag
$$
The condition for its solvability with respect to $g$ reduces to
$$
\begin{equation*}
\frac{\partial}{\partial x_1}\biggl[\beta\frac{\partial a}{\partial x_1}- \gamma a\frac{\partial a}{\partial x_1}\biggr]=\frac{\partial}{\partial x_2} \biggl[\alpha\frac{\partial a}{\partial x_1}- \beta a\frac{\partial a}{\partial x_1}\biggr]=0.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation}
\beta\frac{\partial a}{\partial x_1}-\frac{\gamma}{2}\, \frac{\partial a^2}{\partial x_1}=\xi(x_2).
\end{equation}
\tag{6.3}
$$
Since the left-hand side is independent of $x_2$, we have $\xi=\operatorname{const}$. Averaging both parts of (6.3) over the circle $x_1\, \operatorname{mod} 2\pi$ we get that $\xi=0$. But then
$$
\begin{equation*}
\frac{\partial}{\partial x_1}\biggl[\beta a-\frac{\gamma}{2}a^2\biggr]=0.
\end{equation*}
\notag
$$
Similarly, it follows that Consequently, if $\beta$ or $\gamma$ is not zero, then $a=\operatorname{const}$. This contradicts our assumption. Hence $\beta=\gamma=0$. But then the quadratic integral (6.2) differs from $f_1$ only by a constant factor of $\alpha$.
7. Polynomial integrals of circulatory systemsLet us discuss conditions for the existence of single-valued first integrals polynomial in the momentum for circulatory systems in the restricted sense (in terms of Hodge’s theory; see § 1). More precisely, we consider systems with toric configuration space $\mathbb{T}^2=\{x_1,x_2\, \operatorname{mod} 2\pi\}$ and ‘Euclidean’ kinetic energy $T=(y_1^2+y_2^2)/2$. The force $F$ is represented by a Fourier series
$$
\begin{equation}
\sum f_k e^{i(k,x)},\qquad k \in \mathbb{Z}^2\setminus \{0\},\quad f_k \in \mathbb{C}^2,\quad f_{-k}=\overline{f}_k,
\end{equation}
\tag{7.1}
$$
such that $(f_k,k)=0$ (see § 1 for details). The motion of the circulatory system is described by Newton’s equations (5.1). Let be a polynomial first integral, where $\Phi_j$ is a homogeneous polynomial of degree $j$ in the momentum with coefficients $2\pi$-periodic in $x_1$ and $x_2$. It is easy to show that then the two polynomials $\Phi_m+\Phi_{m-2}+\cdots$ and $\Phi_{m-1}+\Phi_{m-3}+\cdots$ are also first integrals. Indeed, the derivative of $\Phi_j$ with respect to system (5.1) is the sum of two homogeneous forms in $y_1$ and $y_2$ of degrees $j+1$ and $j-1$, respectively. Therefore, the polynomial first integral of degree $m$ should be sought in the formFirst we show that $\Phi_m$ is independent of the angular coordinates. Indeed, the senior homogeneous form is a first integral of the equations of inertial motion on the torus (when $F=0$). Hence
$$
\begin{equation}
\frac{\partial\Phi_m}{\partial x_1}y_1+ \frac{\partial\Phi_m}{\partial x_2}y_2=0.
\end{equation}
\tag{7.3}
$$
We expand $\Phi_m$ in a Fourier series: Substituting into (7.3) we obtain the equality Since $(k,y) \not\equiv 0$ for $k \ne 0$, we have $\varphi_k=0$. So the leading homogeneous form $\Phi_m$ reduces to a function independent of the angular variables $x_1$ and $x_2$. A statement of this kind was actually used by Poincaré in proving the non- integrability of Hamilton’s equations for typical perturbations ([24], Chap. V). By the way, the reasoning that follows has also been inspired by Poincaré’s theory. Now we calculate the derivative of $\Phi$ with respect to (5.1) and equate the terms of degree $m-1$ in the momentum to zero:
$$
\begin{equation}
\biggl(\frac{\partial\Phi_m}{\partial y}\,,F\biggr)+ \biggl(\frac{\partial\Phi_{m-2}}{\partial x}\,,y\biggr)=0.
\end{equation}
\tag{7.4}
$$
Assuming that and considering the Fourier decomposition (7.1), from (7.4) we derive the infinite chain of equalities
$$
\begin{equation}
\biggl(\frac{\partial\Phi_m}{\partial y}\,,f_k\biggr)+ i(k,y)\varphi'_k=0,\qquad k \ne 0.
\end{equation}
\tag{7.5}
$$
Our further analysis is based on the use of these relations. A line $(k,y)=0$, $k \ne 0$, on the momentum plane $\mathbb{R}^2=\{y_1,y_2\}$ is called resonant if $f_k \ne 0$. Hence we have the following. Theorem 3. Let there be $m+1$ different resonant lines. Then any polynomial integral of degree $\leqslant m$ of the circulatory system is a constant. Proof. Consider a straight line $l$ on the momentum plane that intersects $m+1$ resonant lines. The restriction of the homogeneous polynomial $y \mapsto \Phi_m(y)$ to this line is a polynomial of one variable of degree $\leqslant m$. According to the lemma above, this polynomial has at least $m+1$ different zeros. Consequently, it is identically zero. Varying the position of $l$ we obtain $\Phi_m(y) \equiv 0$. Then we consider the first integral $\Phi_{m-2}+\Phi_{m-4}+\cdots$, and in exactly the same way we deduce that $\Phi_{m-2} \equiv 0$. Repeating this process we arrive at the conclusion that $\Phi \equiv 0$ if $m$ is odd. If $m$ is even, then the integral (7.2) reduces to the function $\Phi_0$, which does not depend on the momentum. But then it is obvious that $\Phi_0=\operatorname{const}$. $\Box$ Corollary. If the circulatory system admits a single-valued polynomial integral of degree $m$, then the number of different resonant lines does not exceed $m$. For $m=1$ the converse is also true. Indeed, in this case where $(f_{nk},k)=0$ for all $n \in \mathbb{N}$. But then the equations have the first integral $(k,\dot{x})$, which is linear in theocity.Another example of a circulatory system is of interest. It has two resonant lines and admits a quadratic first integral: Here $f$ and $g$ are $2\pi$-periodic functions of one variable with zero mean values. This system is circulatory in the sense of Hodge’s theory (of course, provided that the force field is non-trivial). The quadratic integral is $\dot{x}_1 \dot{x}_2+W(x_1,x_2)$, whereA direct consequence of Theorem 3 is as follows. Theorem 4. If there is an infinite number of different resonant lines, then the equations of the circulatory system do not admit single-valued non-constant first integrals which are polynomial in the momentum. In particular, a typical circulatory system cannot be reduced to a Hamiltonian system. However, if there is a quadratic integral with non-degenerate quadratic part, then equations (5.1) reduce to the canonical Hamilton equations [8]. In particular, the system of equations (7.6) is Hamiltonian. It remains to prove the lemma. If we put $(k,y)=0$ in (7.4), then we obtain that the vectors $\partial\Phi_m/\partial y$ and $f_k$ are orthogonal. However (because of the circulatory property), $(f_k,k)= 0$. Hence the vector $\partial\Phi_m/\partial y$ is orthogonal to the momentum:
$$
\begin{equation}
\biggl(\frac{\partial\Phi_m}{\partial y}\,,y\biggr)=0
\end{equation}
\tag{7.7}
$$
at points in the resonant line $(k,y)=0$. On the other hand (by Euler’s theorem on homogeneous functions) the left-hand side of (7.7) is $m\Phi_m$. Which is what was required.
8. Circulatory systems with quadratic integralsThe example of a circulatory system (7.6) with two resonant lines and a quadratic integral can be generalized and refined. Note first that the equations of motion of a mechanical system with toric configuration space and ‘standard’ kinetic energy $T=(y_1^2+y_2^2)/2$ under the action of the circulatory force (7.1) can be represented in the following equivalent form:
$$
\begin{equation}
\ddot{x}_1=\frac{\partial W}{\partial x_2}\,,\qquad \ddot{x}_2=-\frac{\partial W}{\partial x_1}\,,
\end{equation}
\tag{8.1}
$$
where $W(x_1,x_2)$ is a smooth function which is $2\pi$-periodic in its arguments. Let us represent $W$ as a Fourier series:
$$
\begin{equation}
W=\sideset{}{'}\sum w_{k}e^{i(k,x)},\qquad k \in \mathbb{Z}^2.
\end{equation}
\tag{8.2}
$$
Resonant lines from § 7, which play a key role in the problem of polynomial integrals of equations (8.1), can be represented in the following form:
$$
\begin{equation*}
\{y \in\mathbb{R}^2\colon (y,k)=0,\ k \ne 0;\ w_k \ne 0\}.
\end{equation*}
\notag
$$
According to Theorem 3, if equations (8.1) admit a first integral quadratic in the momentum, then there are at most two different resonant lines. The converse result also holds. Namely, let there be only two resonant lines, where $p$, $q$ and $r$, $s$ are pairs of mutually prime integers. Then the function $W$ can be represented as a sum where $f(\,\cdot\,)$ and $g(\,\cdot\,)$ are smooth $2\pi$-periodic functions of one variable. They should be assumed to be non-constant; otherwise the straight lines (8.3) are not resonant (the corresponding Fourier coefficients in (8.2) are zero).Theorem 5. Let there be only two resonant lines (8.3). Then the following hold: 1) equations (8.1) admit the integral where
$$
\begin{equation*}
B=\begin{bmatrix} 2pr & ps+qr \\ ps+qr & 2qs \end{bmatrix};
\end{equation*}
\notag
$$
2) if, in addition, $\det B=(ps-qr)^2 \ne 0$, then equations (8.1) are represented in the canonical Hamiltonian form
$$
\begin{equation}
\dot{x}=\frac{\partial H}{\partial u}\,,\quad \dot{u}=-\frac{\partial H}{\partial x},\qquad x \in \mathbb{T}^2,\quad u \in \mathbb{R}^2,
\end{equation}
\tag{8.5}
$$
where 3) if, in addition, the periodic functions $f$ and $g$ are not constant, then equations (8.1) admit no single-valued polynomial integrals that are functionally independent of the quadratic integral (8.4). This statement contains in fact the theory of integrability of circulatory systems (in the restricted sense) that admit a first integral quadratic in the momentum. The condition $\det B\ne 0$ means that the lines (8.3) are different. If $\det B=0$, then the integral (8.4) is the square of a first integral linear in the momentum. The additional conditions in parts 2) and 3) are essential. If they are not satisfied, then equations (8.1) admit two independent integrals and are therefore integrable by the Euler–Jacobi theorem. Indeed, let there be only one resonant line (then $r=p$ and $s=q$). In this case equations (8.1) take the following form: Since $p$ and $q$ are mutually prime, there are integers $u$ and $v$ such that $pv-qu=1$ (by Bezout’s theorem). Let us perform a linear change of angular variables These formulae define an automorphism of the two-dimensional configuration torus. In the new variables equations (8.7) reduce to Hence the functions $\dot{z}_1$ and $\dot{z}_1\dot{z}_2+f(z_1)$ are independent first integrals. The case when $f$ or $g$ takes constant values is considered similarly. Proof of Theorem 5. 1) We verify by direct calculation that the quadratic in the velocities function
$$
\begin{equation}
(p\dot{x}_1+q\dot{x}_2)(r\dot{x}_1+s\dot{x}_2)+(ps-qr)(f-g)
\end{equation}
\tag{8.8}
$$
is a first integral of the system of differential equations (8.1) (in which $W=f+g$). It is, of course, the same as (8.4). The homogeneous quadratic part of the integral (8.8) is zero on the resonant lines (8.3) (as Theorem 3 prescribes).
2) Recall that Lagrange’s equations
$$
\begin{equation}
\frac{d}{dt}\,\frac{\partial L}{\partial\dot{x}}- \frac{\partial L}{\partial x}=0
\end{equation}
\tag{8.9}
$$
admit the Jacobi integral (generalized energy integral)
$$
\begin{equation*}
\biggl(\frac{\partial L}{\partial\dot{x}}\,,\dot{x}\biggr)-L.
\end{equation*}
\notag
$$
In particular, if $L=L_2+L_0$ ($L_k$ is a homogeneous form of degree $k$ in the velocities), then the Jacobi integral is $L_2-L_0$.
Now set
$$
\begin{equation*}
L_2=\frac{1}{2}(B\dot{x},\dot{x})\quad\text{and}\quad L_0=-\det B\,(f-g).
\end{equation*}
\notag
$$
It is not difficult to check that, provided that $\det B \ne 0$, Lagrange’s equations (8.9) with Lagrangian $L_2+L_0$ are equivalent to the original system of equations (8.1) (in which, of course, $W=f+g$). To obtain the canonical Hamiltonian form it remains to apply the classical Legendre transform to the Lagrangian $L_2+L_0$ with non-degenerate quadratic part $L_2$. Thus, from Lagrange’s equations (8.9) we go over to Hamilton’s equations (8.5) with Hamiltonian (8.6).
3) The homogeneous quadratic part of the first integral (8.4) defines a pseudo- Euclidean metric in the velocity space: the matrix $B$ is non-singular and $\det B<0$. Applying the Legendre transform $\dot{x}\mapsto u=B\dot{x}$ we obtain that the quadratic part of the Hamiltonian (8.6) gives the same pseudo-Euclidean metric, but in the dual space. The vectors
$$
\begin{equation*}
\xi=\begin{bmatrix} p \\ q \end{bmatrix}\quad\text{and}\quad \eta=\begin{bmatrix} r \\ s \end{bmatrix}
\end{equation*}
\notag
$$
defining the resonant lines belong to the dual space. It is easy to check that they are isotropic: However, they are not orthogonal with respect to the pseudo-Euclidean metric:
Thus, the Hamiltonian system (8.5) with Hamiltonian (8.6) has the following properties. The resonance set consists of only two lines (their equations in the canonical coordinates have the form $(\xi,B^{-1} u)=0$ and $(\eta,B^{-1} u)=0$), and these lines are not orthogonal in the pseudo-Euclidean metric defined by the quadratic part of the Hamiltonian. Hence (as proved in [3], § 4, Proposition 2, using perturbation theory) under these conditions the canonical Hamiltonian equations do not admit non-trivial single-valued first integrals which are polynomial in the momentum and independent of the Hamiltonian function. 9. Some problems$1^\circ$. It is desirable to develop a theory of integration of the equations of motion of mechanical systems in non-potential force fields taking the particulars of their form into account (as, for example, in the theory of full integrability in Hamiltonian mechanics). The Euler–Jacobi–Lie theorem indicates one possible way. $2^\circ$. It is of interest to extend the result in § 3 to the multidimensional case where the kinetic energy has a Liouville form. There is also the question of whether a more complex separation of variables is possible in the equations of motion of circulatory systems (for instance, in the Stekel coordinates). $3^\circ$. In § 4 an example of a system in a non-potential force field was presented whose equations of motion have a multidimensional invariant torus without quasi- periodic motions. Is such a situation possible in circulatory systems that are integrable by the classical Euler–Jacobi theorem about the last multiplier? The answer is apparently in the affirmative. $4^\circ$. It is desirable to extend the arguments in § 5 on the construction of phase flows in cases (B) and (C) (from § 2) to general dynamical systems satisfying the assumptions of the EJL theorem (in which, by the way, not only Abelian symmetry groups are considered, but also more general nilpotent groups [18]). Some general results about nilpotent flows can be found in [32]. $5^\circ$. In § 6 we gave an example of a system with two degrees of freedom and a toric position space that admits quadratic and cubic integrals. Are there any examples where the system has only one irreducible integral of degree three in the momentum? Irreducibility means that there are no integrals of lower degree. $6^\circ$. It is desirable to extend the result about the absence of polynomial integrals in strictly circulatory systems (§ 7) to the case of ‘non-Euclidean’ kinetic energy and to multidimensional systems. 
Citation in format AMSBIB
\Bibitem{Koz22}
This publication is cited in the following 6 articles:
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