Symmetry Groups of the Planar Three-Body Problem and Action-Minimizing Trajectories

We consider periodic and quasi-periodic solutions of the three-body problem with homogeneous potential from the point of view of equivariant calculus of variations. First, we show that symmetry groups of the Lagrangian action functional can be reduced to groups in a finite explicitly given list, after a suitable change of coordinates. Then, we show that local symmetric minimizers are always collisionless, without any assumption on the group other than the fact that collisions are not forced by the group itself. Moreover, we describe some properties of the resulting symmetric collisionless minimizers (Lagrange, Euler, Hill-type orbits and Chenciner–Montgomery figure-eights).