Periodic Solutions of the Planar N-Center Problem with topological constraints

TL;DR

This paper proves the existence of collision-free periodic solutions in the planar N-center problem within specific topological classes, using variational methods to handle the constraints.

Contribution

It establishes the existence of periodic solutions with topological constraints in the N-center problem, extending previous results to new classes of solutions.

Findings

Existence of collision-free T-periodic solutions for certain homotopy classes.

Application of calculus of variations to constrained N-center problem.

Overcoming challenges in ensuring minimizers are collision-free.

Abstract

In the planar $N$-center problem, for a non-trivial free homotopy class of the configuration space satisfying certain mild condition, we show that there is at least one collision free $T$-periodic solution for any positive $T.$ We use the direct method of calculus of variations and the main difficulty is to show that minimizers under certain topological constraints are free of collision.