# Keplerian orbits through the Conley-Zehnder index

## TL;DR

This paper provides a homotopy theoretical proof that Keplerian ellipses are action minimizers and spectrally stable, using Conley-Zehnder index computations, with potential applications to higher dimensions.

## Contribution

It introduces a homotopy theoretical approach to analyze Keplerian orbits via Conley-Zehnder index, extending classical variational methods to higher dimensions.

## Key findings

- Keplerian ellipses are action minimizers in the plane.
- Keplerian ellipses are spectrally stable as periodic Hamiltonian points.
- The method can be extended to higher-dimensional Keplerian orbits.

## Abstract

It was discovered by Gordon in 1977 that Keplerian ellipses in the plane are minimizers of the Lagrangian action and spectrally stable as periodic points of the associated Hamiltonian flow. The aim of this paper is to give a homotopy theoretical proof of these results through a self-contained, explicit and simple computation of the Conley-Zehnder index. The techniques developed in this paper can be used to investigate the higher dimensional case of Keplerian ellipses, where the classical variational proof no longer applies.