Variational property of periodic Kepler orbits in constant curvature spaces

We study the variational property of the periodic Kepler orbits on the sphere, the plane and the hyperbolic plane. We first classify the orbits by the two constants of motion: the energy and the angular momentum. Then, we characterize the local variational property of the closed orbits by computing...

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Bibliographic Details
Published in:Journal of Differential Equations 2019-11, Vol.267 (10), p.5851-5869
Main Authors: Deng, Yanxia, Diacu, Florin, Zhu, Shuqiang
Format: Article
Language:English
Subjects:
Closed orbits
Curved N-body problem
Kepler problem
Maslov-type index
Morse index
Variational method
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Summary:We study the variational property of the periodic Kepler orbits on the sphere, the plane and the hyperbolic plane. We first classify the orbits by the two constants of motion: the energy and the angular momentum. Then, we characterize the local variational property of the closed orbits by computing the Maslov-type indices. Finally, we study the global variational property of the closed orbits. We prove that the closed orbits on the hyperbolic plane minimizes the action among all loops which encircle the attracting center.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2019.06.008