Local minimality properties of circular motions in $1/r^α$ potentials and of the figure-eight solution of the 3-body problem

Abstract

We first take into account variational problems with periodic boundary conditions, and briefly recall some sufficient conditions for a periodic solution of the Euler-Lagrange equation to be either a directional, a weak, or a strong local minimizer. We then apply the theory to circular orbits of the Kepler problem with potentials of type 1/r α , α > 0. By using numerical computations, we show that circular solutions are strong local minimizers for α > 1, while they are saddle points for α ∈ (0, 1). Moreover, we show that for α ∈ (1, 2) the global minimizer of the action over periodic curves with degree 2 with respect to the origin could be achieved on non-collision and non-circular solutions. After, we take into account the figure-eight solution of the 3-body problem, and we show that it is a strong local minimizer over a particular set of symmetric periodic loops.