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Main results:

1. The Kirchhoff equations are systematically studied using the methods developed by Poincare and Kozlov. Theorems on non-integrability and existence of periodic solutions are proved; stochastization scenarios of the equations of motion are studied.
2. A new integrable problem of non-holonomic mechanics (a problem of motion of a Chaplygin’s ball on an unmovable ball) is discovered. New methods of explicit integration the equations that ensue are developed.
3. А countable family of integrable systems with first integrals of arbitrarily high degree (in a system on the algebra so(4)) are indicated. Bifurcations of the general solution and existence of Poisson structures are studied by the Kovalevskaya method (in cooperation with A.V. Tsygvintsev and S.M. Dudoladov).
4. Transition to chaos in the Euler-Poisson equations in cases of general and restricted problems are investigated. It is shown that stochasticity in classical the Euler-Poisson equations can progress according to Fiegenbaum’s scenario and demonstrates the property of universality.
5. New equations in the dynamics of point vortices on a sphere are derived. Integrability of the problem of three vortices is proved. A classification of the vortex algebra is obtained. Methods of reduction of order for the vortex equations are developed. A new class of periodic solutions – choreographies – is studied, as well as scenarios of transition to chaos. Some new results in the field of stability of Thomson’s configurations on a sphere are obtained.
6. A hierarchy of the dynamic behavior of non-holonomic systems is introduced and investigated. New tensor invariants of non-holonomic systems, connected with rolling of rigid bodies, are found. A new dynamic regime in the motion of a rattleback caused by strange attractors in the phase spaces is found. The results obtained are based on a systematic use of numerical modeling and advanced analytical theory.
7. A new approach to integration of Hamiltonian systems connected with a family of consistent Poisson brackets is developed. New L-A pairs for various integrable systems are found. Such systems with forth and third order integrals generalize the classical results (Kirchhoff and Euler-Poisson equations).
8. A new scheme of integration of non-holonomic systems which extends the method of the so-called Chaplygin’s reduction multiplier is developed. Its efficiency is tested on classical and modern problems of non-holonomic mechanics. A new class of problems of non-holonomic mechanics that arises from various approaches to realization of non-holonomic constraints is introduced.
9. А new class of problems at the interface of theoretical mechanics and hydromechanics connected with interaction between rigid bodies and interaction between rigid bodies and vortex structures in an ideal fluid is investigated. Different forms of equations of motion are derived, their Hamiltonian form is studied and integrals of motion and integrable cases are indicated. A qualitative analysis of motion is performed and occurrence of stochasticity is investigated. New forms of equations governing the motion of so-called mass vortices are obtained and studied.
10. Allied problems from the dynamics of point vortices – motion of Kirchhoff vortices and vortex sources are studied by means of qualitative and burification analysis. New methods of investigation of integrability and qualitative analysis of motion are developed.
11. New methods of investigation of multy-particle systems are elaborated; the corresponding L-A pairs and separation of variables for generalized Toda lattices are found. Stochasticity of motion in Dyson’s system is discussed.
12. Chaplygin’s equations of motion of rigid bodies in the gravitational field (both with and without circulation) are studied. Some integrable cases are indicated; non-integrability is proved by using the separatrix splitting method. Asymptotic motions are studied, and some stability issues, based on and developing the results obtained by V.V. Kozlov, are discussed. New types of motion of a rigid body that occur due to instability of the classical “helical” motions are analyzed.
13. Problems of celestial mechanics and dynamics of rigid body in constant curvature spaces are systematically studied. New integrable cases are indicated and studied qualitatively. Reduction of order in the two-body problem is obtained and integrability analysis is performed. New types of motions are found and their stability is studied. Libration points in the three-body problem are investigated.
14. The main results on exact integration of equations of classical mechanics are classified. New historical facts concerned with creation and development of the basic notions and problems of classical mechanics are pointed out.
15. New integrable cases of motion of point vortices on a sphere, in addition to classical results, are shown. Its qualitative analysis is carried out and new types of stationary configurations are found.