Conservation of ‘Moving’ Energy in Nonholonomic Systems with Affine Constraints and Integrability of Spheres on Rotating Surfaces

Abstract

Energy is in general not conserved for mechanical nonholonomic systems with affine constraints. In this article we point out that, nevertheless, in certain cases, there is a modification of the energy that is conserved. Such a function coincides with the energy of the system relative to a different reference frame, in which the constraint is linear. After giving sufficient conditions for this to happen, we point out the role of symmetry in this mechanism. Lastly, we apply these ideas to prove that the motions of a heavy homogeneous solid sphere that rolls inside a convex surface of revolution in uniform rotation about its vertical figure axis, are (at least for certain parameter values and in open regions of the phase space) quasi-periodic on tori of dimension up to three.MSC: 70F25, 37J60, 37J15, 70E18 * This work is part of the research projects Symmetries and integrability of nonholonomic mechanical systems of the University of Padova and PRIN Teorie geometriche e analitiche dei sistemi Hamiltoniani in dimensioni finite e infinite.

The jacobi integral in nonholonomic mechanics

“…The fact that it exists under additional restrictions (such as smallness of the angular velocity of rotation) was predicted in [26]). It follows from the previous reasoning that such restrictions are not essential.…”

“…In the papers [26,27], which have appeared recently, Fassó explores conditions under which a system with inhomogeneous constraints (1.4) admits the energy integral (1.2).…”

“…It follows from the previous reasoning that such restrictions are not essential. The explicit form of the integral was not presented in [26] either. At the same time, the existence of this integral allows the motion to be explored in more detail.…”

“…However, as noted in the recent paper [26], for the equations of motion of the ball on the surface of rotation to be integrable, the paper [19] lacks an analog of the energy integral (called moving energy in [26]). We note that for inhomogeneous constraints the existence of an energy integral is, generally speaking, nonobvious and in the general case there is no energy integral.…”

“…We note that for inhomogeneous constraints the existence of an energy integral is, generally speaking, nonobvious and in the general case there is no energy integral. In [26] an assertion of its existence in the case of small rotational velocities of the surface is formulated.…”