Antonio Elipe

We consider the motion of an asymmetric gyrostat under the attraction of a uniform Newtonian field. It is supposed that the center of mass lies along one of the principal axes of inertia, while a rotor spins around a different axis of inertia. For this problem, we obtain the possible permanent rotations, that is, the equilibria of the system. The Lyapunov stability of these permanent rotations is analyzed by means of the Energy–Casimir method and necessary and sufficient conditions are derived, proving that there exist permanent stable rotations when the gyrostat is oriented in any direction of the space. The geometry of the gyrostat and the value of the gyrostatic momentum are relevant in order to get stable permanent rotations. Moreover, it seems that the necessary conditions are also sufficient, but this fact can only be proved partially.

We consider the problem of stability of equilibrium points in Hamiltonian systems of two degrees of freedom under low order resonances. For resonances of order bigger than two there are several results giving stability conditions, in particular one based on the geometry of the phase flow and a set of invariants. In this paper we show that this geometric criterion is still valid for low order resonances, that is, resonances of order two and resonances of order one. This approach provides necessary stability conditions for both the semisimple and non-semisimple cases, with an appropriate choice of invariants.

This paper reviews the Serret–Andoyer (SA) canonical formalism in rigid-body dynamics, and presents some new results. As is well known, the problem of unsupported and unperturbed rigid rotator can be reduced. The availability of this reduction is offered by the underlying symmetry, that stems from conservation of the angular momentum and rotational kinetic energy. When a perturbation is turned on, these quantities are no longer preserved. Nonetheless, the language of reduced description remains extremely instrumental even in the perturbed case. We describe the canonical reduction performed by the Serret–Andoyer (SA) method, and discuss its applications to attitude dynamics and to the theory of planetary rotation. Specifically, we consider the case of angular-velocity-dependent torques, and discuss the variation-of-parameters-inherent antinomy between canonicity and osculation. Finally, we address the transformation of the Andoyer variables into action-angle ones, using the method of Sadov.

We are interested in studying the properties and perturbations of orbits around a central planet surrounded by a ring. The problem has been studied a long time ago by Laplace, Maxwell and others. Maxwell considered a ring composed of a number of discrete masses orbiting in a circular orbit. Gauss also derived the potential due to a solid circular ring and its derivation is reproduced in Kellog's textbook on Potential Theory. The potential can be evaluated in terms of a complete elliptic integral of the first kind. Computing the accelerations also requires a second kind elliptic integral. We have experimented with at least three different methods for computing the potential and its first partial derivatives: Gauss quadrature, the Carlson functions and the Arithmetico-Geometric mean. The standard formulation breaks down near the center of the ring which is an unstable equilibrium point but a linearization can be made near this point. We have also studied the efficiency of the Spherical Harmonic expansion of the ring potential. This expansion has only the even zonal terms and thus no tesserals.
In a preliminary study, we have looked at planar periodic orbits (and their stability), around a ring without a central body, both in the plane of the ring and the plane orthogonal to it. We find nearly a dozen types and families of periodic orbits. Several of these families seem to end with an orbit that collides with the ring. One of the goals of this preliminary study is to understand the effect of the singularity at the ring itself, (where the potential and the accelerations become infinite).