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- W2020455417 abstract "We prove the existence of a number of smooth periodic motions u ∗ of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group $mathcal{R}$ of one of the five Platonic polyhedra. The number N coincides with the order $|mathcal{R}|$ of $mathcal{R}$ and the particles have all the same mass. Our approach is variational and u ∗ is a minimizer of the Lagrangian action $mathcal{A}$ on a suitable subset $mathcal{K}$ of the H 1 T-periodic maps u:ℝ→ℝ3N . The set ${mathcal {K}}$ is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group $mathcal{R}$ . There exist infinitely many such cones ${mathcal {K}}$ , all with the property that ${mathcal {A}}|_{{mathcal {K}}}$ is coercive. For a certain number of them, using level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the N-body problem with a rich geometric–kinematic structure." @default.
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- W2020455417 date "2010-12-16" @default.
- W2020455417 modified "2023-09-26" @default.
- W2020455417 title "Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem" @default.
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- W2020455417 doi "https://doi.org/10.1007/s00222-010-0306-3" @default.
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