Action minimizers under topological constraints in the planar equal-mass four-body problem

TL;DR

This paper demonstrates the existence of new collision-free action minimizers in the planar equal-mass four-body problem, connecting symmetric boundary configurations and extending to periodic orbits.

Contribution

It introduces two new classes of action minimizers with specific topological and symmetry constraints, expanding the known solutions in the four-body problem.

Findings

Existence of collision-free minimizers connecting symmetric configurations.

Extension of minimizers to periodic or quasi-periodic orbits.

Application of level estimate method to prove collision-free property.

Abstract

It is shown that in the planar equal-mass four-body problem, there exist two sets of new action minimizers connecting two planar boundary configurations with fixed symmetry axes and specific order constraints on the four bodies: a double isosceles configuration and an isosceles trapezoid configuration. By applying the level estimate method, these minimizers are shown to be collision-free and they can be extended to two new sets of periodic or quasi-periodic orbits.