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Energy is in general not conserved for mechanical nonholonomic systems withaffine constraints. In this article we point out that, nevertheless, in certaincases, there is a modification of the energy that is conserved. Such a functioncoincides with the energy of the system relative to a different referenceframe, in which the constraint is linear. After giving sufficient conditionsfor 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 homogeneoussolid sphere that rolls inside a convex surface of revolution in uniformrotation about its vertical figure axis, are (at least for certain parametervalues and in open regions of the phase space) quasi-periodic on tori ofdimension up to three.

Conservation of `moving' energy in nonholonomic systems with affine constraints and integrability of spheres on rotating surfaces