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| We consider a planar mechanical system consisting of $n$ point particles subject to a potential which is invariant under rigid rotations. We address the question of \emph{linear and spectral stability} for a particular class of periodic solutions termed \emph{relative equilibria}, providing a rather general sufficient condition for their instability.
Notable applications are represented by two generalisations of the $n$\nobreakdash-bo\-dy problem involving two kinds of singular potentials: the $\alpha$-homogene\-ous one, with $\alpha \in (0, 2)$, which includes the gravitational case, and the logarithmic one. More specifically, we focus on the $3$-body-type problem (governed by the aforementioned potentials) and compute the Morse index of the Lagrangian circular orbit and of its iterates by availing ourselves of some symplectic techniques and an index theorem involving the Maslov index.
This is a joint work with Vivina Barutello and Alessandro Portaluri. |
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