Blow-up for realizing homotopy classes in the three-body problem

TL;DR

This paper discusses McGehee blow-up as a key tool in proving that all free homotopy classes in the planar three-body problem can be realized by periodic solutions, introducing new insights and explicit descriptions.

Contribution

It introduces an energy-balance motivated transformation of McGehee blow-up and provides an explicit description of the blown-up phase space for the planar N-body problem.

Findings

Every free homotopy class can be realized by a periodic solution.

Explicit description of the blown-up phase space as a complex vector bundle.

Novel use of energy-balance in McGehee blow-up transformation.

Abstract

This expository note describes McGehee blow-up \cite{McGehee} in its role as one of the main tools in my recent proof with Rick Moeckel \cite{RM2} that every free homotopy class for the planar three-body problem can be realized by a periodic solution. The main novelty is my use of energy-balance to motivate the transformation of McGehee. Another novelty is an explicit description of the blown-up reduced phase space for the planar N-body problem, $N \ge 3$ as a complex vector bundle over the half-line times complex projective $N-2$-space. The half line coordinate is the size of the labelled planar N-gon whose vertices are the instantaneous positions of the N bodies and the projective space coordinatizes the shape of this N-gon body.