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2013
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Francesco Fasso

 Fasso F., Ramos A., Sansonetto N. On the Existence of Invariant Tori in Nearly-Integrable Hamiltonian Systems with Finitely Differentiable Perturbations 2007, vol. 12, no. 6, pp.  579-588 Abstract 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. Keywords: nonholonomic systems, first integrals, first integrals linear in the velocities, symmetries of nonholonomic systems, reaction forces, Noether theorem Citation: Fasso F., Ramos A., Sansonetto N.,  On the Existence of Invariant Tori in Nearly-Integrable Hamiltonian Systems with Finitely Differentiable Perturbations, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 579-588 DOI:10.1134/S1560354707060019
 Benettin G., Fasso F., Guzzo M. Nekhoroshev-stability of $L_4$ and $L_5$ in the spatial restricted three-body problem 1998, vol. 3, no. 3, pp.  56-72 Abstract We show that $L_4$ and $L_5$ in the spatial restricted circular three-body problem are Nekhoroshev-stable for all but a few values of the reduced mass up to the Routh critical value. This result is based on two extensions of previous results on Nekhoroshev-stability of elliptic equilibria, namely to the case of "directional quasi-convexity", a notion introduced here, and to a (non-convex) steep case. We verify that the hypotheses are satisfied for $L_4$ and $L_5$ by means of numerically constructed Birkhoff normal forms. Citation: Benettin G., Fasso F., Guzzo M.,  Nekhoroshev-stability of $L_4$ and $L_5$ in the spatial restricted three-body problem, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 56-72 DOI:10.1070/RD1998v003n03ABEH000080