Francesco Fasso
Publications:
Fasso F., Sansonetto N.
On Some Aspects of the Dynamics of a Ball in a Rotating Surface of Revolution and of the Kasamawashi Art
2022, vol. 27, no. 4, pp. 409423
Abstract
We study some aspects of the dynamics of the nonholonomic system
formed by a heavy homogeneous ball constrained to roll without sliding
on a steadily rotating surface of revolution. First, in the case in
which the figure axis of the surface is vertical (and hence the
system is $\textrm{SO(3)}\times\textrm{SO(2)}$symmetric) and the surface has a
(nondegenerate) maximum at its vertex, we show the existence of
motions asymptotic to the vertex and rule out the possibility of blowup.
This is done by passing to the 5dimensional $\textrm{SO(3)}$reduced system.
The $\textrm{SO(3)}$symmetry persists when the figure axis of the surface is
inclined with respect to the vertical — and the system can be viewed
as a simple model for the Japanese kasamawashi (turning umbrella)
performance art — and in that case we study the (stability of the)
equilibria of the 5dimensional reduced system.

Fasso F., Sansonetto N.
Conservation of Energy and Momenta in Nonholonomic Systems with Affine Constraints
2015, vol. 20, no. 4, pp. 449462
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 "gaugelike" generalizations, in timeindependent nonholonomic mechanical systems with affine constraints. These conditions involve geometrical and mechanical properties of the system, and are codified in the socalled reactionannihilator distribution.

Bates L., Fasso F.
An Affine Model for the Actions in an Integrable System with Monodromy
2007, vol. 12, no. 6, pp. 675679
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.

Fasso F., Ramos A., Sansonetto N.
On the Existence of Invariant Tori in NearlyIntegrable Hamiltonian Systems with Finitely Differentiable Perturbations
2007, vol. 12, no. 6, pp. 579588
Abstract
We consider nonholonomic systems with linear, timeindependent constraints subject to ositional conservative active forces. We identify a distribution on the configuration manifold, that we call the reactionannihilator 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.

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. 117
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.

Fasso F., Giacobbe A., Sansonetto N.
Periodic flows, ranktwo Poisson structures, and nonholonomic mechanics
2005, vol. 10, no. 3, pp. 267284
Abstract
It has been recently observed that certain (reduced) nonholonomic systems are Hamiltonian with respect to a ranktwo 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.

Benettin G., Fasso F., Guzzo M.
Nekhoroshevstability of $L_4$ and $L_5$ in the spatial restricted threebody problem
1998, vol. 3, no. 3, pp. 5672
Abstract
We show that $L_4$ and $L_5$ in the spatial restricted circular threebody problem are Nekhoroshevstable 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 Nekhoroshevstability of elliptic equilibria, namely to the case of "directional quasiconvexity", a notion introduced here, and to a (nonconvex) steep case. We verify that the hypotheses are satisfied for $L_4$ and $L_5$ by means of numerically constructed Birkhoff normal forms.
