First Integrals of Equations of Motion of a Heavy Rotational Symmetric Body on a Perfectly Rough Plane

Summary

We consider the problem of the motion of a heavy dynamically symmetric rigid body bounded by a surface of rotation on a fixed perfectly rough horizontal plane. The integrability of this problem was proved by S.A. Chaplygin [1]. Chaplygin has found that the equations of motion of given mechanical system have, besides the energy integral, two first integrals, linear in generalized velocities. However, the explicit form of these integrals is known only in the case, when the moving body is a nonhomogeneous dynamically symmetric ball. In the case, when the moving body is a round disk or a hoop, the integrals, linear in the velocities, are expressed using hypergeometric series [1],[2],[3]. In the paper of Kh.M. Mushtari [4] the investigation of this problem was continued. For additional restrictions, imposed on the surface of moving body and its mass distribution, Mushtari has found two particular cases, when the motion of the body can be investigated completely. In the first case the moving rigid body is bounded by a surface formed by rotation of an arc of a parabola about the axis, passing through its focus, and in the second case, the moving body is a paraboloid of rotation. For other bodies, bounded by a surface of rotation and moving without sliding on a horizontal plane, the explicit form of additional first integrals is unknown.

In this work we find some new cases when all the integrals of the problem can be expressed explicitly, when the surface of moving body satisfies to a Mushtari condition. The set of surfaces of moving bodies satisfying to this condition is described.