Entire minimal parabolic trajectories: the planar anisotropic Kepler problem

TL;DR

This paper analyzes minimal parabolic trajectories in the planar anisotropic Kepler problem using a variational approach, characterizing their existence, linking it to collision avoidance, and exploring related periodic orbits.

Contribution

It extends the variational analysis of parabolic orbits to the planar anisotropic case, characterizing all such orbits as Morse minimizers and connecting their existence to potential collisions.

Findings

Parabolic orbits are characterized as Morse minimizers in a homotopy class.

Existence of parabolic trajectories is linked to the absence of collisions.

Periodic trajectories with nontrivial homotopy type are related to the same threshold.

Abstract

We continue the variational approach to parabolic trajectories introduced in our previous paper [5], which sees parabolic orbits as minimal phase transitions. We deepen and complete the analysis in the planar case for homogeneous singular potentials. We characterize all parabolic orbits connecting two minimal central configurations as free-time Morse minimizers (in a given homotopy class of paths). These may occur for at most one value of the homogeneity exponent. In addition, we link this threshold of existence of parabolic trajectories with the absence of collisions for all the minimizers of fixed-ends problems. Also the existence of action minimizing periodic trajectories with nontrivial homotopy type can be related with the same threshold.