| Contents |
| We consider the $N$-body problem with interaction potential
$U_\alpha=\frac{1}{\vert x_i-x_j\vert^\alpha}$ for $\alpha>1$. We assume
that the particles have all the same mass and that $N$ is the
order $\vert\mathcal{R}\vert$ of the rotation group $\mathcal{R}$ of one
of the five Platonic polyhedra. We study motions that, up to a relabeling
of the $N$ particles, are invariant under
$\mathcal{R}$. By variational techniques we prove the existence of periodic
and chaotic motions. |
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