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.
['Francesco Fassò', 'Nicola Sansonetto']
Conservation of `moving' energy in nonholonomic systems with affine constraints and integrability of spheres on rotating surfaces