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

DOI10.1007/S42985-022-00148-5arXiv2201.01205WikidataQ114216449 ScholiaQ114216449MaRDI QIDQ6387482

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Publication date: 4 January 2022

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 $1/r^{a}lpha,,alpha>0$. By using numerical computations, we show that circular solutions are strong local minimizers for $alpha>1$, while they are saddle points for $alphain(0,1)$. Moreover, we show that for $alphain(1,2)$ 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.

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