Local minimality properties of circular motions in 1/rα potentials and of the figure-eight solution of the 3-body problem

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(alpha),alpha >0. By using numerical computations, we show that circular solutions are strong local minimizers for alpha >1, while they are saddle points for alpha is an element of(0,1). Moreover, we show that for alpha is an element of(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.