# Index Theory for Zero Energy Solutions of the Planar Anisotropic Kepler Problem

## TL;DR

This paper introduces a method to compute Morse indices of zero energy solutions in the planar anisotropic Kepler problem, revealing a link between these indices and the solutions' oscillatory behavior, with implications for variational analysis.

## Contribution

It provides a novel approach to calculating Morse indices for zero energy solutions, connecting them to oscillation patterns in the anisotropic Kepler problem.

## Key findings

- Morse indices can be computed explicitly for these solutions.
- A connection between Morse indices and oscillatory behavior is established.
- The method simplifies analysis of singular Lagrange systems.

## Abstract

In the variational study of singular Lagrange systems, the zero energy solutions play an important role. In this paper we find a simple way of computing the Morse indices of these solutions for the planar anisotropic Kepler problem. In particular an interesting connection between the Morse indices and the oscillating behaviors of these solutions discovered by the physicist M. Gutzwiller is established.