text: 244,49,1484259890939969539,1202973751316537348,Marco Fenucci,"📢📢 New #paper available on Partial Differential Equations and Applications, by @SpringerNature! This is the fourth paper extracted from my #PhD thesis, and a preprint is freely available on @arxiv #AcademicTwitter <LINK> <LINK> This work has been possible also thanks to the support from @Stardust_H2020 #MSCA",https://arxiv.org/abs/2201.01205,"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 $\alpha \in (0,1)$. Moreover, we show that for $\alpha \in (1,2)$ 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. ","Local minimality properties of circular motions in $1/r^\alpha$