Francesco Fassò

Lie series method for vector fields and Hamiltonian perturbation theory

We consider a rigorous Hamiltonian perturbation theory based on the transformation of the vector field of the system, realized by the Lie method. Such a perturbative technique presents some advantages over the standard one, which uses the transformation of the Hamilton functions. Indeed, the present method is simple, and furnishes quite detailed informations on the normal form. Moreover, it leads to estimates which are better and/or simpler than those of the scalar Lie methods. The perturbation method is presented with reference to two model problems, both pertaining to the realm of the well known Nekhoroshev theorem: the confining of actions for exponentially long times in a system of coupled harmonic oscillators, and an application to the so called problem of the realization of a holonomic constraint in classical mechanics.

Geometry of KAM tori for nearly integrable Hamiltonian systems

We obtain a global version of the Hamiltonian KAM theorem for invariant Lagrangian tori by gluing together local KAM conjugacies with the help of a partition of unity. In this way we find a global Whitney smooth conjugacy between a nearly integrable system and an integrable one. This leads to the preservation of geometry, which allows us to define all non-trivial geometric invariants of an integrable Hamiltonian system (like monodromy) for a nearly integrable one.

The Euler-Poinsot top: a non-commutatively integrable system without global action-angle coordinates

We study the global structure of the fibration by the invariant two-dimensional tori of the Euler-Poinsot top — the rigid body with a fixed point and no torques. We base our analysis on the notion of bifibration (or dual pair) which, as results from the approach based on the so-called non-commutative integrability, provides a thorough description of the geometry of integrable degenerate Hamiltonian systems. In this way, we get a global geometric picture of the Euler-Poinsot system which fully accounts for its degeneracy through the (Poisson) structure of the base of the fibration by the two-dimensional invariant tori. In particular, we explain in this way why this system does not possess global ‘generalized’ action-angle coordinates: the obstructions are the topological non-triviality of the fibration by the invariant two-dimensional tori and the compactness of the symplectic leaves of its base manifold. We also compare this description with the usual description based on the notion of complete integrability, and we remark that, as a general fact, such a common approach fails to provide a natural, thorough description of degenerate systems.

The Exact Computation of the Free Rigid Body Motion and Its Use in Splitting Methods

This article investigates the use of the computation of the exact free rigid body motion as a component of splitting methods for rigid bodies subject to external forces. We review various matrix and quaternion representations of the solution of the free rigid body equation which involve Jacobi ellipic functions and elliptic integrals and are amenable to numerical computations. We consider implementations which are exact (i.e., computed to machine precision) and semiexact (i.e., approximated via quadrature formulas). We perform a set of extensive numerical comparisons with state-of-the-art geometrical integrators for rigid bodies, such as the preprocessed discrete Moser-Veselov method. Our numerical simulations indicate that these techniques, combined with splitting methods, can be favorably applied to the numerical integration of torqued rigid bodies.

A Changing-Chart Symplectic Algorithm for Rigid Bodies and Other Hamiltonian Systems on Manifolds

We revive the elementary idea of constructing symplectic integrators for Hamiltonian flows on manifolds by covering the manifold with the charts of an atlas, implementing the algorithm in each chart (thus using coordinates) and switching among the charts whenever a coordinate singularity is approached. We show that this program can be implemented successfully by using a splitting algorithm if the Hamiltonian is the sum H1+H2 of two (or more) integrable Hamiltonians. Profiting from integrability, we compute exactly the flows of H1 and H2 in each chart and thus compute the splitting algorithm on the manifold by means of its representative in any chart. This produces a symplectic algorithm on the manifold which possesses an interpolating Hamiltonian, and hence it has excellent properties of conservation of energy. We exemplify the method for a point constrained to the sphere and for a symmetric rigid body under the influence of positional potential forces.

Stability Properties of the Riemann Ellipsoids

Abstract The Riemann ellipsoids are steady motions of an ideal, incompressible, self-gravitating fluid that retain an ellipsoidal shape. The existence, properties, and stability of these steady motions have been investigated since Newtons time. Most of the stability results are due to Riemann, who studied Lyapunov stability, and to Chandrasekhar, who studied (primarily numerically) spectral stability, thus obtaining Lyapunov instability results. This article addresses the “Nekhoroshev stability” (stability for finite, but very long time scales) of those Riemann ellipsoids that are spectrally stable but of unknown Lyapunov stability. We base our analysis on a Hamiltonian formulation of the problem derived from Riemanns original formulation (which we interpret here as a formulation on a covering space) using recent results from Hamiltonian perturbation theory. Given the complexity of the system, we resort to numerical calculations at certain steps of the stability analysis. As a prerequisite to our analysis, we repeat the Lyapunov and spectral stability analyses, finding important discrepancies with Chandrasekhars findings. We provide numerical evidence that(i) There are spectrally stable ellipsoids of type II and the region of spectral stability of the ellipsoids of type III is significantly larger than that found by Chandrasekhar. The regions of spectral stability of the ellipsoids of types I, II and III have a finer and subtler structure than was previously believed.(ii) All Riemann ellipsoids, except a finite number of codimension-one resonant subfamilies, are Nekhoroshev-stable.

Fast rotations of the rigid body: a study by Hamiltonian perturbation theory. Part II: Gyroscopic rotations

We continue the analysis which began in part 1 of this paper of the long-time behaviour of the fast rotations of a rigid body in an external analytic force field. Specifically, we consider the motions of a symmetric rigid body whose angular velocity is nearly parallel to the symmetry axis of the ellipsoid of inertia, which were excluded from the previous analysis because of the singularity of the action-angle coordinates. By suitably implementing the techniques of Nekhoroshevs theorem, so as to overcome this difficulty, we provide a description of these motions on timescales which grow with the angular velocity as ; such a description confirms the general properties of fast motions established in part 1.

The reaction-annihilator distribution and the nonholonomic Noether theorem for lifted actions

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°, 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°. Since the fibers of ℜ° 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.

Comparison of splitting algorithms for the rigid body

We compare several different second-order splitting algorithms for the asymmetric rigid body, with the aim of determining which one produces the smallest energy error for a given rigid body, namely, for given moments of inertia. The investigation is based on the analysis of the dominant term of the modified Hamiltonian and indicates that different algorithms can produce energy errors which differ by several orders of magnitude. As a byproduct of this analysis we remark that, for the special case of a flat rigid body with moments of inertia proportional to (1, 0.75, 0.25), one of the considered algorithms is in fact of order four.

For Hamiltonian perturbation theory to symplectic

Hamiltonian perturbation theory explains how symplectic integrators work and, in particular, why they can be used to measure extremely small energy exchanges between different degrees of freedom in molecular collision problems. Conversely, numerical experiments based on symplectic integrators permit a detailed understanding of the dynamics of nearly integrable Hamiltonian systems, thus providing a valuable support to Hamiltonian perturbation theory.