Morse Index for Homothetic motions in the gravitational n-body problem

TL;DR

This paper establishes a clear dichotomy for the Morse index of homothetic motions in the gravitational n-body problem, showing it is either zero or infinite depending on a non-spiraling condition, regardless of the configuration's index or energy level.

Contribution

It proves that the Morse index for homothetic motions is either zero or infinite based solely on the non-spiraling condition, extending previous results to all such motions.

Findings

Morse index is zero if the non-spiraling condition holds.

Morse index is infinite if the non-spiraling condition does not hold.

The result is independent of the size of the central configuration's index or energy level.

Abstract

In the gravitation n-body Problem, a homothetic orbit is a special solution of the Newton's Equations of motion, in which each body moves along a straight line through the center of mass and forming at any time a central configuration. In 2020, Portaluri et al. proved that under a spectral gap condition on the limiting central configuration, known in literature as non-spiraling or [BS]-condition, the Morse index of an asymptotic colliding motion is finite. Later Ou et al. proved this result for other classes of unbounded motions, e.g. doubly asymptotic motions (e.g. doubly homothetic motions). In this paper we prove that for a homothetic motion, irrespective of how large the index of the limiting central configuration and how large the energy level is, the following alternative holds: if the non-spiraling condition holds then the Morse index is 0 otherwise it is infinite.