Regularizable collinear periodic solutions in the $n$-body problem with arbitrary masses

TL;DR

This paper proves the existence of regularizable collinear periodic solutions in the $n$-body problem with arbitrary positive masses, including symmetric cases and a new proof for the Schubart orbit when $n=3$.

Contribution

It establishes the existence of regularizable collinear periodic solutions for any mass ordering and provides a new proof of the Schubart orbit for three bodies.

Findings

Existence of collinear periodic solutions for arbitrary mass arrangements.

Solutions transition from binary collision to another within half a period.

Additional symmetry in solutions when mass equality conditions are met.

Abstract

For $n$-body problem with arbitrary positive masses, we prove there are regularizable collinear periodic solutions for any ordering of the masses, going from a simultaneous binary collision to another in half of a period with half of the masses moving monotonically to the right and the other half monotonically to the left. When the masses satisfy certain equality condition, the solutions have extra symmetry. This also gives a new proof of the existence of Schubart orbit, when $n=3$.