Periodic orbits of the planar anisotropic Kepler problem

Agraïments: The second author of this work was partially supported by Fundación Séneca de la Región de Murcia grant number 19219/PI/14.In this paper we prove that at every energy level the anisotropic problem with small anisotropy has two periodic orbits which bifurcate from elliptic orbits of the Kepler problem with high eccentricity. Moreover we provide approximate analytic expressions for these periodic orbits. The tool for proving this result is the averaging theory

Periodic orbits of the planar anisotropic generalized Kepler problem

Many generalizations of the Kepler problem with homogeneous potential of degree -1/2 have been considered. Here, we deal with the generalized anisotropic Kepler problem with homogeneous potential of degree -1. We provide the explicit solutions of this problem on the zero energy level and show that all of them are periodic

Symmetric Periodic Solutions of the Anisotropic Manev Problem

We consider the Manev Potential in an anisotropic space, i.e., such that the force acts differently in each direction. Using a generalization of the Poincare' continuation method we study the existence of periodic solutions for weak anisotropy. In particular we find that the symmetric periodic orbits of the Manev system are perturbed to periodic orbits in the anisotropic problem.Comment: Late

Singularity in classical and quantum Kepler Problem with Weak Anisotropy

Anisotropic Kepler problem is investigated by perturbation method in both classical and quantum mechanics. In classical mechanics, due to the singularity of the potential, global diffusion in phase space occurs at an arbitrarily small perturbation parameter. In quantum mechanics, the singularity induces a large transition amplitude between quasi degenerate eigen states, which generically decays as $\hbar$ in the semi-classical limit.Comment: 6 pages, 2 figures, 1 tabl

Nonintegrability and Chaos in the Anisotropic Manev Problem

The anisotropic Manev problem, which lies at the intersection of classical, quantum, and relativity physics, describes the motion of two point masses in an anisotropic space under the influence of a Newtonian force-law with a relativistic correction term. Using an extension of the Poincare'-Melnikov method, we first prove that for weak anisotropy, chaos shows up on the zero-energy manifold. Then we put into the evidence a class of isolated periodic orbits and show that the system is nonintegrable. Finally, using the geodesic deviation approach, we prove the existence of a large non-chaotic set of uniformly bounded and collisionless solutions

Periodic orbits of the two fixed centers problem with a variational gravitational field

Within a given range of energy levels the two fixed centers problem under a variational gravitational field admits periodic orbits bifurcating from the Kepler problem. The analytical expressions of these periodic orbits are given when the mass parameter of the system is sufficiently small

On the Periodic Orbits of the Perturbed Two- and Three-Body Problems

In this work, a perturbed system of the restricted three-body problem is derived when the perturbation forces are conservative alongside the corresponding mean motion of two primaries bodies. Thus, we have proved that the first and second types of periodic orbits of the rotating Kepler problem can persist for all perturbed two-body and circular restricted three-body problems when the perturbation forces are conservative or the perturbed motion has its own extended Jacobian integral

Periodic orbits of the planar anisotropic Manev problem and of the perturbed hydrogen atom problem

In this paper we use the averaging theory for studying the periodic solutions of the planar anisotropic Manev problem and of two perturbations of the hydrogen atom problem. When a convenient parameter is sufficiently small we prove that for every value e∈ (0, 1) a unique elliptic periodic solution with eccentricity e of the Kepler problem can be continued to the mentioned three problems