Local minimality properties of circular motions in $$1/r^\alpha $$ 1 / r α 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 $$1/r^\alpha , \, \alpha > 0$$ 1 / r α , α > 0 . By using numerical computations, we show that circular solutions are strong local minimizers for $$\alpha > 1$$ α > 1 , while they are saddle points for $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) . Moreover, we show that for $$\alpha \in (1,2)$$ 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.