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2013
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# Francesco Fasso

University of Padova

## Publications:

 Fasso F., Sansonetto N. Conservation of Energy and Momenta in Nonholonomic Systems with Affine Constraints 2015, vol. 20, no. 4, pp.  449-462 Abstract 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. Keywords: nonholonomic mechanical systems, conservation of energy, reaction-annihilator distribution, gauge momenta, nonholonomic Noether theorem Citation: Fasso F., Sansonetto N.,  Conservation of Energy and Momenta in Nonholonomic Systems with Affine Constraints, Regular and Chaotic Dynamics, 2015, vol. 20, no. 4, pp. 449-462 DOI:10.1134/S1560354715040048
 Bates L., Fasso F. An Affine Model for the Actions in an Integrable System with Monodromy 2007, vol. 12, no. 6, pp.  675-679 Abstract We give a standard model for the flat affine geometry defined by the local action variables of a completely integrable system. We are primarily interested in the affine structure in the neighborhood of a critical value with nontrivial monodromy. Keywords: monodromy, completely integrable systems Citation: Bates L., Fasso F.,  An Affine Model for the Actions in an Integrable System with Monodromy, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 675-679 DOI:10.1134/S1560354707060093
 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. Long term stability of proper rotations and local chaotic motions in the perturbed Euler rigid body 2006, vol. 11, no. 1, pp.  1-17 Abstract The long term stability of the proper rotations of the perturbed Euler rigid body was recently investigated analytically in the framework of Nekhoroshev theory. In this paper we perform a parallel numerical investigation, with the double aim of illustrating the theory and to submit it to a critical test. We focus the attention on the case of resonant motions, for which the stability is not trivial (resonant proper rotations are indeed stable in spite of the presence of local chaotic motions, with positive Lyapunov exponent, around them). The numerical results indicate that the analytic results are essentially optimal, apart from a particular resonance, actually the lowest order one, where the system turns out to be more stable than the theoretical expectation. Keywords: rigid body, Nekhoroshev theory, chaotic dynamics Citation: Benettin G., Fasso F., Guzzo M.,  Long term stability of proper rotations and local chaotic motions in the perturbed Euler rigid body , Regular and Chaotic Dynamics, 2006, vol. 11, no. 1, pp. 1-17 DOI: 10.1070/RD2006v011n01ABEH000331
 Fasso F., Giacobbe A., Sansonetto N. Periodic flows, rank-two Poisson structures, and nonholonomic mechanics 2005, vol. 10, no. 3, pp.  267-284 Abstract It has been recently observed that certain (reduced) nonholonomic systems are Hamiltonian with respect to a rank-two Poisson structure. We link the existence of these structures to a dynamical property of the (reduced) system: its periodicity, with positive period depending continuously on the initial data. Moreover, we show that there are in fact infinitely many such Poisson structures and we classify them. We illustrate the situation on the sample case of a heavy ball rolling on a surface of revolution. Keywords: Poisson structures, non-holonomic systems, periodic flows Citation: Fasso F., Giacobbe A., Sansonetto N.,  Periodic flows, rank-two Poisson structures, and nonholonomic mechanics , Regular and Chaotic Dynamics, 2005, vol. 10, no. 3, pp. 267-284 DOI:10.1070/RD2005v010n03ABEH000315
 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