The questions of justification of the Gibbs canonical distribution for systems with elastic impacts are discussed. A special attention is paid to the description of probability measures with densities depending on the system energy.

A time-periodic Hamiltonian system on a cotangent bundle of a compact manifold with Hamiltonian strictly convex and superlinear in the momentum is studied. A hyperbolic Diophantine nondegenerate invariant torus $N$ is said to be minimal if it is a Peierls set in the sense of the Aubry–Mather theory. We prove that $N$ has an infinite number of homoclinic orbits. For any family of homoclinic orbits the first and the last intersection point with the boundary of a tubular neighborhood $U$ of $N$ define sets in $U$. If there exists a compact family of minimal homoclinics defining contractible sets in $U$, we obtain an infinite number of multibump homoclinic, periodic and chaotic orbits. The proof is based on a combination of variational methods of Mather and a generalization of Shilnikov's lemma.

It is well-known that in real-analytic multi-frequency slow-fast ODE systems the dependence of the right-hand sides on fast angular variables can be reduced to an exponentially small order by a near-identical change of the variables. Realistic constructive estimates for the corresponding exponentially small terms are obtained.

In the modern approach to integrable Hamiltonian systems, their representation in the Lax form (the Lax pair or the $L$–$A$ pair) plays a key role. Such a representation also makes it possible to construct and solve multi-dimensional integrable generalizations of various problems of dynamics. The best known examples are the generalizations of Euler's and Clebsch's classical systems in the rigid body dynamics, whose Lax pairs were found by Manakov [10] and Perelomov [12]. These Lax pairs include an additional (spectral) parameter defined on the compactified complex plane or an elliptic curve (Riemann surface of genus one). Until now there were no examples of $L$–$A$ pairs representing physical systems with a spectral parameter running through an algebraic curve of genus more than one (the conditions for the existence of such Lax pairs were studied in [11]).
In the given paper we consider a new Lax pair for the multidimensional Manakov system on the Lie algebra $so(m)$ with a spectral parameter defined on a certain unramified covering of a hyperelliptic curve. An analogous $L$–$A$ pair for the Clebsch–Perelomov system on the Lie algebra $e(n)$ can be indicated.
In addition, the hyperelliptic Lax pair enables us to obtain the multidimensional generalizations of the classical integrable Steklov–Lyapunov systems in the problem of a rigid body motion in an ideal fluid. The latter is known to be a Hamiltonian system on the algebra $e(3)$. It turns out that these generalized systems are defined not on the algebra $e(n)$, as one might expect, but on a certain product $so(m)+so(m)$. A proof of the integrability of the systems is based on the method proposed in [1].

We study the $C^r$-convergence of the compositions $W_n=U_1U_2\cdots U_n$ where mappings $U_k$ tend to the identity transformation in the $C^r$-topology as $k \to\infty$. The cases $r = 0$ and $1 \leqslant r < +\infty$ turn out to be drastically different.

In this work stability of polygonal configurations on a plane and sphere is investigated. The conditions of linear stability are obtained. A nonlinear analysis of the problem is made with the help of Birkhoff normalization. Some problems are also formulated.

The motion of a point over an $n$-dimensional nondegenerate quadric in one-dimentioned quadratic potential under the assumption that there exist $n+1$ mutually orthogonal planes of symmetry is considered. It is established, that all cases of the existence of an algebraic complete commutative set of integrals are exhausted by classical ones. The question whether the integrability due to Liouville is inherited by invariant symplectic submanifolds is studied. In algebraic category for submanifolds of dimension $4$ such integrability is valid.

A modified method of A.A. Slutskin (1963) of analytical extension to the complex time plane of solutions of a single-frequency nonlinear Hamiltonian system with slowly varying parameters is considered. On the basis of this method a proof of the estimate for the accuracy of persistence of adiabatic invariant due to A.A. Slutskin is given for such systems.

It is known that the problem of an investigation of invariant sets (in particular stationary motions) of mechanical systems with symmetries can be reduced to the problem of the analysis of the effective potential [1-11]. The effective potential represents the minimum of the total mechanical energy with respect to quasivelocities on fixed levels of Noether's integrals corresponding to symmetries of the system. The effective potential is a function in the configuration space depending on constants of Noether's integrals. This function is defined in such points of the configuration space where Noether's integrals independent and can have singularities at some points where these integrals are dependent.

The absence of an additional meromorphic first integral of a Hamiltonian system with two degrees of freedom emerging in describing of the Friedman cosmological models with the coupled scalar field is proved.

We study perturbations of Hamiltonian systems of $n+1$ degrees of freedom $(n \geqslant 2)$ in the real-analytic case, such that in the absence of the perturbation they contain a partially hyperbolic (whiskered) $n$-torus with the Kronecker flow on it with a Diophantine frequency, connected to itself by a homoclinic exact Lagrangian submanifold (separatrix), formed by the coinciding unstable and stable manifolds (whiskers) of the torus. Typically, a perturbation causes the separatrix to split. We study this phenomenon as an application of the version of the KAM theorem, proved in [13]. The theorem yields the representations of global perturbed separatrices as exact Lagrangian submanifolds in the phase space. This approach naturally leads to a geometrically meaningful definition of the splitting distance, as the gradient of a scalar function on a subset of the configuration space, which satisfies a first order linear homogeneous PDE. Once this fact has been established, we adopt a simple analytic argument, developed in [15] in order to put the corresponding vector field into a normal form, convenient for further analysis of the splitting distance. As a consequence, we argue that in the systems, which are Normal forms near simple resonances for the perturbations of integrable systems in the action-angle variables, the splitting is exponentially small.