An Index Theory for Collision, Parabolic and Hyperbolic Solutions of the Newtonian $n$-body Problem

TL;DR

This paper develops an index theory for specific solutions of the Newtonian $n$-body problem, linking Morse indices to spectral properties and providing tools for analyzing collision and infinity-bound solutions.

Contribution

It introduces a new index theory for collision, parabolic, and hyperbolic solutions, relating Morse indices to spectral data of central configurations.

Findings

Morse and Maslov indices for solutions with arbitrary energy are characterized.

A formula relating Morse indices of homothetic solutions to spectra of the potential is derived.

Results may aid in applying non-action minimization methods to the $n$-body problem.

Abstract

In the Newtonian $n$-body problem for solutions with arbitrary energy, which start and end either at a total collision or a parabolic/hyperbolic infinity, we prove some basic results about their Morse and Maslov indices. Moreover for homothetic solutions with arbitrary energy, we give a simple and precise formula that relates the Morse indices of these homothetic solutions to the spectra of the normalized potential at the corresponding central configurations. Potentially these results could be useful in the application of non-action minimization methods in the Newtonian $n$-body problem.