On the reduction of some systems of classical mechanics to the Liouvillian form

DOI: 10.34759/trd-2023-128-02

Аuthors

Kuleshov A. S.

Lomonosov Moscow State University, 1, Leninskie Gory, Moscow, 119991, Russia

e-mail: kuleshov@mech.math.msu.su

Abstract

In 1846, J. Liouville indicated a class of holonomic mechanical systems for which the Hamilton — Jacobi equation can be integrated by the method of separation of variables. Mechanical systems belonging to this class are called Liouvillian systems. In the case of two degrees of freedom, it is known, that the Liouvillian system admits, in addition to the energy integral, another first integral, quadratic in generalized velocities. The presence of two first integrals for a holonomic mechanical system with two degrees of freedom allows us to assert that such a system is integrable. Moreover, it is possible to study the bifurcations of the joint levels of the first integrals. The corresponding method for Liouvillian systems with two degrees of freedom was developed by Ya. V. Tatarinov and V. M. Alekseev. To use this method, it is necessary to reduce the studied mechanical system to the Liouvillian form.

The paper considers several well-known systems of classical mechanics (the Euler — Poinsot case of the problem of the motion of a rigid body about a fixed point, the Jacobi problem of geodesics on an ellipsoid, the problem of motion of the Chaplygin sphere on the perfectly rough horizontal plane), which, by changing variables, take the form of Liouvillian systems with two degrees of freedom. Moreover, to reduce the problem of rolling motion of the Chaplygin sphere to the Liouvillian system, the theory of Chaplygin reducing multiplier is used. Previously this approach was not used in solving the Chaplygin sphere rolling problem. As a result, the study of the dynamics of the considered systems can be carried out using the methods of the theory of topological analysis with the separation of variables according to Liouville.

Keywords:

Liouvillian system, Euler – Poinsot problem, Jacobi problem, Chaplygin sphere

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