Periodic orbits of Hamiltonian systems: Applications to perturbed Kepler problems

Abstract

Abstract. We provide for a class of Hamiltonian systems in the actionangle variables sufficient conditions for showing the existence of periodic orbits. We expand this result to the study of the existence of periodic orbits of perturbed spatial Keplerian Hamiltonians with axial symmetry. Finally, we apply these general results for finding periodic orbits of the MateseWhitman Hamiltonian, of the spatial anisotropic Hamiltonian and of the spatial generalized van der Waals Hamiltonian.

Periodic solutions of symmetric Kepler perturbations and applications

“…In our work we obtain periodic solutions of Hamiltonian function which are perturbation of the integrable Kepler problem with 3 degrees of freedom. Our results combine the discrete symmetries of the Hamiltonian and the Continuation Poincaré method, using strongly the first approximation of the solutions of the full Hamiltonian system given by a variational system, although these ideas have been used in other works (see for example, [1], [2], [4], [9], [16], [20], [21], [22], [23], etc.) for specific perturbations or under a different point of view.…”

“…See also [9]. Of course, this Hamiltonian function H MW H in (11.1) is a particular case of the Hamiltonian (1.1) and is invariant under the anti-symplectic reflections S 1 and S 2 .…”

Improving the averaging theory for computing periodic solutions of the differential equations

“…The averaging theory for computing periodic solutions of a differential system is one of the best analytical tools for the study of the periodic solutions, see for instance the papers [23,13,19,12,14,17,18,20,10,11,8].…”

Periodic Orbits of Third Kind in the Zonal J2 + J3 Problem

“…In the following, we shall use the Delaunay variables for studying the periodic orbits of the Hamiltonian system associated with the Hamiltonian (1), which is called the zonal J 2 + J 3 problem; see [13][14][15][16][17][18] for more details on the Delaunay variables and this astrophysical problem. Thus, if {l, g, k, L, G, K} are the action angle coordinates of Delaunay, where l is the mean anomaly, g is the argument of the periapsis of the unperturbed elliptical orbit measured in the invariant plane, k is the longitude of the ascending node, L is the square root of the major semi-axis of the unperturbed elliptic orbit, G is the modulus of the total angular momentum, and K is the third component of the angular momentum, then the Hamiltonian (1) has the form:…”

“…The main aim of the present work is to study sufficient conditions for the existence of periodic orbits of the third kind in the J 2 + J 3 zonal problem. For this purpose, we are inspired by the methodology developed in [13], and explain in detail its most relevant aspects in Appendix A. Finally, we state the existence of three families of polar periodic orbits and the existence of four families of inclined periodic orbits.…”