New families of symmetric periodic solutions of the spatial anisotropic Manev problem

Abstract

Applying perturbation theory and symmetry conditions, we prove the existence of new families of periodic orbits for the 3-dimensional anisotropic Manev problem, which depends on three parameters.

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.…”

“…In order to enunciate our main results, under our approach, we consider conveniently modified Poincaré-Delaunay variables (see details for example in [1], [3], [15], [17], [21]). In fact, the main reasons to consider these types of variables can be summarized as follows: Firstly, in these coordinates the characterization of "reflection" symmetries is simpler, secondly because the periodicity equation (equation that characterizes the initial conditions of symmetric periodic solutions) can be reduced to a minimum number of equations, and in third place the elimination of the degeneracy due to periodicity (maximal rank).…”