Effective Hamiltonian for the D'Alembert Planetary Model Near a Spin/Orbit Resonance |
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| Authors: | Luca Biasco Luigi Chierchia |
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| Institution: | (1) SISSA/ISAS, t, 34013 Trieste, Italy;(2) Dipartimento di Matematica, Università 'Roma Tre', Largo S. L. Murialdo 1, 00146 Roma, Italy |
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| Abstract: | The D'Alembert model for the spin/orbit problem in celestial mechanics is considered. Using a Hamiltonian formalism, it is shown that in a small neighborhood of a p:q spin/orbit resonance with (p,q) different from (1,1) and (2,1) the 'effective' D'Alembert Hamiltonian is a completely integrable system with phase space foliated by maximal invariant curves; instead, in a small neighborhood of a p:q spin/orbit resonance with (p,q) equal to (1,1) or (2,1) the 'effective' D'Alembert Hamiltonian has a phase portrait similar to that of the standard pendulum (elliptic and hyperbolic equilibria, separatrices, invariant curves of different homotopy). A fast averaging with respect to the 'mean anomaly' is also performed (by means of Nekhoroshev techniques) showing that, up to exponentially small terms, the resonant D'Alembert Hamiltonian is described by a two-degrees-of-freedom, properly degenerate Hamiltonian having the lowest order terms corresponding to the 'effective' Hamiltonian mentioned above. |
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| Keywords: | Hamiltonian systems D'Alembert model spin/orbit resonances fast averaging Nekhoroshev normal forms proper degeneracies stability effective Hamiltonian |
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