# Free time minimizers for the planar three-body problem

## TL;DR

This paper characterizes free time minimizers in the planar three-body problem, showing they are asymptotic to Lagrange's solution and identifying conditions on mass ratios that influence their behavior.

## Contribution

It proves the asymptotic behavior of solutions to the three-body problem related to free time minimizers and characterizes the influence of mass ratios on these solutions.

## Key findings

- Solutions asymptotic to Lagrange's solution are free time minimizers.
- Every free time minimizer tends to Lagrange's solution under certain mass ratios.
- Excluded asymptotic to Euler configurations using second variation arguments.

## Abstract

Free time minimizers of the action (called"semi-static" solutions by Ma\~ne) play a central role in the theory of weak KAM solutions to the Hamilton-Jacobi equation (see Fathi). We prove that any solution to Newton's three-body problem which is asymptotic to Lagrange's parabolic homothetic solution is eventually a free time minimizer. Conversely, we prove that every free time minimizer tends to Lagrange's solution, provided the mass ratios lie in a certain large open set of mass ratios. We were inspired by the work of Da Luz-Maderna who had shown that every free time minimizer for the N-body problem is parabolic, and therefore must be asymptotic to the set of central configurations. We exclude being asymptotic to Euler's central configurations by a second variation argument. Central configurations correspond to rest points for the McGehee blown-up dynamics. The large open set of mass ratios are those for which the linearized dynamics at each Euler rest point has a complex eigenvalue.