Solutions homoclinic to regular precessions of a symmetric satellite

The aim of this paper is to study numerically asymptotic manifolds and homoclinic solutions to the regular precessions of a rigid symmetric satellite in a circular orbit.     

Non-integrability of a Weakly Integrable Hamiltonian System

The geometric approach to mechanics based on the Jacobi metric allows to easily construct natural mechanical systems which are integrable (actually separable) at a fixed value of the energy. The aim of the present paper is to investigate the dynamics of a simple prototype system outside the zero-energy hypersurface. We find that the general situation is that in which integrability is not preserved at arbitrary values of the energy. The structure of the Hamiltonian in the separating coordinates at zero energy allows a perturbation treatment of this system at energies slightly different from zero, by which we obtain an analytical proof of non-integrability.This revised version was published online in October 2005 with corrections to the Cover Date.     

Bruns' Theorem: The Proof and Some Generalizations

We give here a proof of Bruns’ Theorem which is both complete and as general as possible: Generalized Bruns’ Theorem.In the Newtonian (n+1)-body problem in ℝ ^p with n≥2 and 1≤p≤n+1, every first integral which is algebraic with respect to positions, linear momenta and time, is an algebraic function of the classical first integrals: the energy, the p(p−1)/2 components of angular momentum and the 2p integrals that come from the uniform linear motion of the center of mass. Bruns’ Theorem only dealt with the Newtonian three-body problem in ℝ³; we have generalized the proof to n+1 bodies in ℝ^p with p≤n+1. The whole proof is much more rigorous than the previous versions (Bruns, Painlevé, Forsyth, Whittaker and Hagiara). Poincaré had picked out a mistake in the proof; we have understood and developed Poincaré’s instructions in order to correct this point (see Subsection 3.1). We have added a new paragraph on time dependence which fills in an up to now unnoticed mistake (see Section 6). We also wrote a complete proof of a relation which was wrongly considered as obvious (see Section 3.3). Lastly, the generalization, obvious in some parts, sometimes needed significant modifications, especially for the case p=1 (see Section 4). This revised version was published online in July 2006 with corrections to the Cover Date.     

Non-Integrability and Structure of the Resonance Zones in a Class of Galactic Potentials

The structure of the resonance zone in nearly integrable Hamiltonian systems is studied by a more general method than the pendulum approximation. This method applies to the case of a non-degenerate integrable part in the Hamiltonian. This problem may be overcome in a class of galactic-type polynomial potentials, in the case where the higher-order term is by itself integrable. An illustrative example is worked out.     

On the measure of the structure around the last kam torus before and after its break-up

We review theorems for proving non-integrability of Hamiltonian dynamical systems, which are based on properties of the variational equations in real or complex time or on the destruction of the resonant tori of an integrable system under a perturbation.     

Non—integrability of Hill's lunar problem

We prove that Hill's lunar problem does not possess a second analytic integral of motion, independent of the Hamiltonian. In order to obtain this result, we avoid the usual normalization in which the angular velocity of the rotating reference frame is put equal to unit. We construct an artificial Hamiltonian that includes an arbitrary parameter b and show that this Hamiltonian does not possess an analytic integral of motion for in an open interval around zero. Then, by selecting suitable values of , b and using the invariance of the Hamiltonian under scaling in the units of length and time, we show that the Hamiltonian of Hill's problem does not possess an integral of motion, analytically continued from the integrable two–body problem in a rotating frame.