Abstract
We study the dynamics of an elastic body whose shape and position evolve due to the gravitational forces exerted by a pointlike planet. The main result is that, if all the deformations of the satellite dissipate some energy, then under a suitable nondegeneracy condition there are only three possible outcomes for the dynamics: (i) the orbit of the satellite is unbounded, (ii) the satellite falls on the planet, (iii) the satellite is captured in synchronous resonance i.e. its orbit is asymptotic to a motion in which the barycenter moves on a circular orbit, and the satellite moves rigidly, always showing the same face to the planet. The result is obtained by making use of LaSalle’s invariance principle and by a careful kinematic analysis showing that energy stops dissipating only on synchronous orbits. We also use in quite an extensive way the fact that conservative elastodynamics is a Hamiltonian system invariant under the action of the rotation group.
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This research was founded by the Prin project 2010–2011 “Teorie geometriche e analitiche dei sistemi Hamiltoniani in dimensioni finite e infinite”.
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Haus, E., Bambusi, D. Asymptotic Behavior of an Elastic Satellite with Internal Friction. Math Phys Anal Geom 18, 14 (2015). https://doi.org/10.1007/s11040-015-9184-7
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DOI: https://doi.org/10.1007/s11040-015-9184-7