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

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 is the pull-back of the energy of the system written in a system of time-dependent coordinates in which the constraint is linear, and for this reason will be called a 'moving' energy. After giving sufficient conditions for the existence of a conserved, time-independent moving energy, 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.