Analysis of the Hess integrable case in the problem of motion of a ball on a smooth plane


Аuthors

Kuleshov A. S.*, Lobanova E. V.**

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

*e-mail: kuleshov@mech.math.msu.su
**e-mail: yorik0603helena@gmail.com

Abstract

The problem of motion of a heavy nonhomogeneous ball on a fixed smooth horizontal plane is considered. This problem is similar in many aspects to the classical problem of motion of a heavy rigid body with a fixed point. Both of these problems can be represented in the form of Hamiltonian system with two degrees of freedom. For integrability of both of these problems, only one additional first integral is needed. The equations of motion of both problems have the energy integral and the area integral. There are known integrable cases in the problem of motion of a ball, similar to the Euler – Poinsot, Lagrange and Hess integrable cases in the problem of motion of a heavy rigid body with a fixed point. In this paper, we study the integrable case of the problem of motion of the ball, similar to the Hess case. Equations of motion of the ball are written using the special coordinate system, originally introduced by P.V. Kharlamov to study various problems of rigid body dynamics. The study showed that the equations of motion of a nonhomogeneous ball on a smooth plane in the Hess case and the equations of motion of a heavy rigid body with a fixed point in the Hess case have similar properties. In particular, it is shown that, as in the classical problem of the motion of a body with a fixed point, a qualitative description of the motion of a ball on a smooth horizontal plane is reduced to the integration of the second – order linear differential equation, and at the zero level of the area integral the equations of motion of the ball can be integrated in quadratures.

Keywords:

Motion of a Nonhomogeneous Ball, Smooth Plane, Hess integrable case

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