Positive energy solutions in the anisotropic Kepler problem with homogeneous potential

TL;DR

This paper investigates positive energy solutions in the anisotropic Kepler problem with homogeneous potential, establishing their asymptotic properties and proving the existence of hyperbolic and bi-hyperbolic solutions with specified behaviors.

Contribution

It introduces new existence results for hyperbolic and bi-hyperbolic solutions with prescribed asymptotic behaviors in the anisotropic Kepler problem.

Findings

Asymptotic properties of positive energy solutions are characterized.

Existence of hyperbolic solutions with given initial and asymptotic conditions is proved.

Bi-hyperbolic solutions with specified behaviors are established in the planar case.

Abstract

We study positive energy solutions of the anisotropic Kepler problem with homogeneous potential. First some asymptotic property of positive energy solutions is obtained, as time goes to infinity. Afterwards, we prove the existence of hyperbolic solutions with given initial configuration and asymptotic behavior, when time goes to positive or negative infinity, and in the planar case, the existence of bi-hyperbolic solutions with given asymptotic behaviors, when time goes to both positive and negative infinities, under various conditions.