Symplectic coordinates on S²×S² for perturbed Keplerian problems: Application to the dynamics of a generalised Størmer problem

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• 作者：Manuel Iñ ; arrea ; V&#xed ; ctor Lanchares ; Jesú ; s F. Palaci&#xe1 ; n ; Ana I. Pascual ; J. Pablo Salas ; Patricia Yanguas
• 关键词：Product of two 2-spheres ; Symplectic reduction ; Perturbed Keplerian problems ; Delaunay coordinates ; Poincar&#xe9 ; -like coordinates ; Stø ; rmer problem ; Periodic solutions ; KAM tori
• 刊名：Journal of Differential Equations
• 出版年：2011
• 出版时间：1 February 2011
• 年：2011
• 卷：250
• 期：3
• 页码：1386-1407
• 全文大小：260 K

文摘

In order to analyse the dynamics of a given Hamiltonian system in the space defined as the Cartesian product of two spheres, we propose to combine Delaunay coordinates with Poincaré-like coordinates. The coordinates are of local character and have to be selected accordingly with the type of motions one has to take into consideration, so we distinguish the following types: (i) rectilinear motions; (ii) circular and equatorial motions; (iii) circular non-equatorial motions; (iv) non-circular equatorial motions; and (v) non-circular and non-equatorial motions. We apply the theory to study the dynamics of the reduced flow of a generalised Størmer problem that is modelled as a perturbation of the Kepler problem. After using averaging and reduction theories, the corresponding flow is analysed on the manifold S²×S², calculating the occurring equilibria and their stability. Finally, the flow of the original problem is reconstructed, concluding the existence of some families of periodic solutions and KAM tori.