Hamiltonian methods in stability and bifurcations problems for artificial satellite dynamics
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The dynamics of a rigid body in a central gravitational eld can be modelled by a Hamiltonian system with continuous symmetries implemented by an action of the group SO(3). Depending on the particular geometry of the body (as for instance if the body is axisymmetric) this symmetry group can even be enlarged. There are many classical studies of steadily rotating solutions of this system based on various approximate models of the orbital-attitude coupling of arti cial Earth satellites, but these models don't fully exploit the geometric structure of the problem. [WKM90] provides a geometrical description of this problem and studies its relative equilibria (steady solutions) using Poisson reduction and the Energy-Cassimir method. Our approach consists in attacking this problem by means of the more recent Reduced-Energy-Momentum [SLM91], since it clari es and improves the existing results and, furthermore, it allows for a systematic bifurcation analysis. One novelty of this work with respect to previous approaches is that we also treat axisymmetric bodies using this geometric formalism. Employing these techniques we are able to explicitly and previously unknown relative equilibria and to study their stability and bifurcation patterns.