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We first take into account variational problems with periodic boundaryconditions, and briefly recall some sufficient conditions for a periodicsolution of the Euler-Lagrange equation to be either a directional, a weak, ora strong local minimizer. We then apply the theory to circular orbits of theKepler problem with potentials of type $1/r^\alpha, \, \alpha > 0$. By usingnumerical computations, we show that circular solutions are strong localminimizers for $\alpha > 1$, while they are saddle points for $\alpha \in(0,1)$. Moreover, we show that for $\alpha \in (1,2)$ the global minimizer ofthe action over periodic curves with degree $2$ with respect to the origincould be achieved on non-collision and non-circular solutions. After, we takeinto account the figure-eight solution of the 3-body problem, and we show thatit is a strong local minimizer over a particular set of symmetric periodicloops.

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