# An Index theory for asymptotic motions under singular potentials

## TL;DR

This paper introduces an index theory for asymptotic solutions in the n-body problem, providing conditions for spectral index finiteness and a Maslov-type index to analyze trajectories with collapse or escape behaviors.

## Contribution

It develops a novel index framework for parabolic and collision solutions, handling non-compactness and asymptotic behaviors in the classical n-body problem.

## Key findings

- Finiteness conditions for spectral index of asymptotic motions.
- A Maslov-type index suitable for half-clinic orbits.
- A relative index theory capturing the Hamiltonian structure.

## Abstract

We develop an index theory for parabolic and collision solutions to the classical n-body problem and we prove sufficient conditions for the finiteness of the spectral index valid in a large class of trajectories ending with a total collapse or expanding with vanishing limiting velocities. Both problems suffer from a lack of compactness and can be brought in a similar form of a Lagrangian System on the half time line by a regularising change of coordinates which preserve the Lagrangian structure. We then introduce a Maslov-type index which is suitable to capture the asymptotic nature of these trajectories as half-clinic orbits: by taking into account the underlying Hamiltonian structure we define the appropriate notion of geometric index for this class of solutions and we develop the relative index theory.