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.
We consider nonholonomic systems with linear, time-independent constraints subject to positional conservative active forces. We identify a distribution on the configuration manifold, that we call the reaction-annihilator distribution R degrees, the fibers of which are the annihilators of the set of all values taken by the reaction forces on the fibers of the constraint distribution. We show that this distribution, which can be effectively computed in specific cases, plays a central role in the study of first integrals linear in the velocities of this class of nonholonomic systems. In particular we prove that, if the Lagrangian is invariant under (the lift of) a group action in the configuration manifold, then an infinitesimal generator of this action has a conserved momentum if and only if it is a section of the distribution R degrees. Since the fibers of R degrees contain those of the constraint distribution, this version of the nonholonomic Noether theorem accounts for more conserved momenta than what was known so far. Some examples are given
We characterize the conditions for the conservation of the energy and of the components of the momentum maps of lifted actions, and of their "gauge-like" generalizations, in time-independent nonholonomic mechanical systems with affine constraints. These conditions involve geometrical and mechanical properties of the system, and are codified in the so-called reaction-annihilator distribution.
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