On the Existence of Invariant Tori in Nearly-Integrable Hamiltonian Systems with Finitely Differentiable Perturbations

We consider nonholonomic systems with linear, time-independent constraints subject to ositional conservative active forces. We identify a distribution on the configuration manifold, that we call the reaction-annihilator distribution $\mathcal{R}^\circ$, 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 $\mathcal{R}^\circ$. Since the fibers of $\mathcal{R}^\circ$ contain those of the constraint distribution, this version of the nonholonomic Noether theorem accounts for more conserved omenta than what was known so far. Some examples are given.