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Main results:

Rigid body dynamics

a) Solution of the Poincare problem about non-existance of new analytical integrals in the problem of rotation of a heavy asymmetrical rigid body with a fixed point. The existence of infinite number of non-degenerate long period trajectories and a stochastic area near splitted separatrices.

b) The classical results due to Steklov and Lyapunov on integrability of Kirchhoff's equations of motion of rigid body in a fluid are expanded.

c) Solution of the Chaplygin problem on falling motion of a heavy rigid body in an unbounded volume of ideal fluid.

d) Auto-rotational modes in the falling motion of a heavy body in a viscous fluid are found and investigated..

e) Qualitative analysis of Kovalevskaya's and the Goryachev-Chaplygin gyroscope.

f) Discovery of controllability of a body with rigid shell and varying distribution of its mass.

Integrability of dynamical equations

а) Solution of the Painlive-Golubev problem on the connection between bifurcation of solutions on the plane of complex time and the existence of a complete set of holomorphic first integrals of Hamilton's equations.

b) Theorems on topological obstacles for existence of first integrals of dynamical equations and relation between the degree of irreducible polynomial integrals and the topology of the configuration space.

c) Discovery of stochastic behaviour in the system of particles that interact with a smooth periodic potential. Classification of completely integrable generalized Toda chains.

Theory of stability

а) A complete and rigorous proof of the fundamental theorem due to Irnshow (1839) about instability of an equilibrium in the force filed with harmonic potential.

b) Development and adaptation of the first Lyapunov method to highly non-linear systems.

c) Discovery of the relation between the degree of instability and the index of inertia of Lyapunov functions.

d) Description of the spectrum of a linear system with quadratic integral with the help of the symplectic geometry of Artin's complex space.

Variational methods

а) Variational proof of existence of periodic trajectories and estimation of their number depending on the topological invariants of domains of possible motion.

b) Application of the Hamilton principle to determination of solutions which are asymptotic to equilibria of non-autonomous Lagrangian systems.

Fundamental principles of dynamics

а) Development of a constructive method in the dynamics of systems with constraints. New mathematical models for systems with unilateral and non-integrable constraints.

b) Discovery of an integral analogy to the Gauss principle.

c) Establishing of connection between the dynamics on invariant manifolds of Hamiltonian equations and the hydrodynamics of ideal fluid which led to creation of a so-called vortex approach to integration of canonical Hamiltonian equations..

Nonholonomic mechanics

а) Proof of non-existence of invariant measure for a typical nonholonomic system.

b) Improvement of some methods for integration of nonholonomic systems.

Impact theory

а) Pass to the limit in impact systems when a unilateral constraint is replaced by Kelvin-Voight medium whose stiffness and viscosity coefficient go to infinity. On this basis, a new effective method for stability analysis of impact-vibrating systems is developed.

b) Discovery of new completely integrable systems of billiard type.

c) Connection between extremal characteristics of two-link billiard trajectories and their stability.

Symmetry and integral invariants

а) Obstacles to existence of non-trivial symmetry groups are established. These are: destruction of invariant resonant tori and complicated topology of the configuration space.

b) Connection between the phase-space symmetries of a system with two degrees of freedom and multivalued first integrals.

c) Proof of the Poincare hypothesis about nonexistence of new integral invariants in the restricted three-body problem.

Statistical mechanics

а) The canonical Gibbs distribution for systems with finite number of degrees of freedom is obtained without resort to the ergodic hypothesis.

b) Development of the kinetic of a collisionless continuous medium that occupies a region with rigid or slowly changing boundary.

c) Investigation of weak limits of solutions to the Louisville equation for non-linear Hamiltonian systems and conditions for rough entropy to increase.

d) Creation of reversible non-equilibrium statistical mechanics for Boltzmann-Gibbs gas.

Ergodic theory

а) Investigation of the final properties of integrals of quasi-periodic functions.

b) Discovery of uniform recurrence in dynamical systems with integral invariant on a torus and the limiting mixture in such systems.

c) Generalization of the extended law of large numbers using the Riss and Voronoy convergence.

d) Proof a new variation of the ergodic theorem for the case when time average is replaced by parameter average (say, the total energy) with fixed density.

Mathematical physics

а) Discovery of the Lagrange turbulence in typical stationary flows of viscous fluid.

b) Description of conservation laws in quantum systems (which arise from polynomial differential operators) with toroidal configuration space.

c) Determination of non-trivial solutions to the Klein-Gordon equations in the de Sitter space with finite action.