Constructing invariant tori for the spatial Hill lunar problem

Abstract

In this paper, the spatial Hill lunar problem is investigated, and the existence of invariant tori of hyperbolic type in a neighborhood of its equilibrium is shown. Moreover, the author checks the non-degenerate condition analytically and obtains two-dimensional elliptic invariant tori on its central manifold as well.

Keywords

Spatial Hill lunar problem / KAM theorem / Quasi-periodic orbits / Elliptic invariant tori

References

[1] Arnold V. I.. Small denominators and problems of stability of motion in classical and celestial mechanics. Usp. Mat. Nauk, 1963, 18(6): 91-192

[2] Arnold V. I.. Engl. Transl.. Russ. Math. Surv, 1963, 18(6): 85-191

[3] Arnold V. I., Kozlov V. V., Neishtadt A. I.. Mathematical Aspects of Classical and Celestial Mechanics, 2006, Berlin: Springer-Verlag

[4] Birkhoff G. D.. Dynamical Systems, 1927, Providence, RI: Amer. Math. Soc.

[5] Carr J.. Applications of Center Manifold Theory, 1981, Berlin: Springer-Verlag

[6] Chauvineau, B. and Mignard, F., ATLAS of the Circular Planar Hills Problem, Observatoire de la Cote dAzur, CERGA, 1991, 1–71.

[7] Delshams A., Masdemont J., Roldán P.. Computing the scattering map in the spatial Hill’s problem. Discrete Contin. Dyn. Syst. Ser. B, 2008, 10(2–3): 455-483

[8] Gómez G., Marcote M., Mondelo J. M.. The invariant manifold structure of the spatial Hill’s problem. Dyn. Syst, 2005, 20: 115-147

[9] Guckenheimer J., Holmes P.. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 1991, New York: Springer-Verlag

[10] Hénon M.. Numerical exploration of the restricted problem, V, Hills case: Periodic orbits and their stability. Astronomy and Astrophysics, 1969, 1: 223-238

[11] Hénon M.. Numerical exploration of the restricted problem, VI, Hills case: Non-periodic orbits. Astronomy and Astrophysics, 1970, 9: 24-36

[12] Hénon M.. Vertical stability of periodic orbits in the restricted problem, Hills case. Astronomy and Astrophysics, 1974, 30: 317-321

[13] Laub A., Meyer K. R.. Canonical forms for symplectic and Hamiltonian matrices. Celest. Mech, 1974, 9: 213-238

[14] Mielke A.. Hamiltonian and Lagrangian Flows on Center Manifolds with Applications to Elliptic Variational Problems, 1991, Berlin: Springer-Verlag

[15] Moser J.. Stable and Random Motions in Dynamical Systems with Special Emphasis on Celestial Mechanics, 2001, Princeton: Princeton University Press

[16] Moser J.. Lectures on Hamiltonian Systems, Mem. Amer. Math. Soc., 1968, Providence, RI: American Mathematical Society

[17] Vashkovyak M. A., Teslenko N. M.. On periodically evolving orbits in the single averaged Hill problem. Astronomy Letters, 2008, 34: 280-288

[18] Villac B. F.. Dynamics in the Hill problem with applications to spacecraft maneuvers, 2008, Ann Arbor, Michigan: University of Michigan

[19] Voyatzis G., Gkolias I., Varvoglis H.. The dynamics of the elliptic Hill problem: Periodic orbits and stability regions. Celestial Mechanics and Dynamical Astronomy, 2012, 113(1): 125-139

[20] Wintner A.. The Analytic Foundations of Celestial Mechanics, 1941, Princeton: Princeton University Press