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| In the last decades a zoo of new symmetric periodic collisionless orbits for the $N$\nobreakdash-bo\-dy problem appeared in the
literature as minimizers of the Lagrangian action functional. Certainly one of the key features of such orbits is
their Morse index, as well as their linear (in)stability properties. A central device useful to deeply understand
these questions is based on a well-known symplectic invariant: the Maslov index.
In this talk we revise some important achievements in the field, stressing the importance of a Bott-type
iteration formula for periodic solutions which are the object of a finite group action, and we present some interesting
perspectives necessary to try to penetrate the intricate dynamics of the problem. |
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