From the circular to the spatial elliptic restricted three-body problem
We deal with the study of the spatial restricted three-body problem in the case where the small particle is far from the primaries, that is, the so-called comet case. We consider the circular problem, apply double averaging and compute the relative equilibria of the reduced system. It appears that, in the circular problem, we find not only part of the equilibria existing in the elliptic case, but also new ones. These critical points are in correspondence with periodic and quasiperiodic orbits and invariant tori of the non-averaged Hamiltonian. We explain carefully the transition between the circular and the elliptic problems. Moreover, from the relative equilibria of elliptic type, we obtain invariant 3-tori of the original system.
KeywordsSpatial restricted three-body problem Averaging Relative equilibria Stability Bifurcations Invariant tori
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- Abraham, R., Marsden, J.E.: Foundations of Mechanics. Dover, Redwood City, CA (1985)Google Scholar
- Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics. Dynamical Systems. III, Encyclopaedia Mathematics Science, 3. Springer-Verlag, Berlin (1997)Google Scholar
- Brouwer, D., Clemence, G.M.: Methods of Celestial Mechanics. Academic Press, New York, London (1961)Google Scholar
- Meyer, K.R., Hall, G.R.: Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. Applied Mathematical Sciences 90. Springer-Verlag, New York (1992)Google Scholar
- Palacián, J.F., Yanguas, P.: Invariant manifolds of spatial restricted three-body problems: the lunar case. In: Delgado, J. Lacomba, E.A., Llibre J., Pérez-Chavela E. (eds.) New Advances in Celestial Mechanics and Hamiltonian Systems, pp. 199–221. Kluwer Academic/Plenum Publishers, Dordrecht (2004)Google Scholar
- Szebehely, V.: Theory of Orbits. Academic Press, New York (1967)Google Scholar