  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.
