Inventiones mathematicae

, Volume 185, Issue 2, pp 283–332 | Cite as

Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem



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 H1T-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.


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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Dipartimento di Matematica Pura ed ApplicataUniversità di L’AquilaL’AquilaItaly
  2. 2.Dipartimento di MatematicaUniversità di PisaPisaItaly
  3. 3.Dipartimento di MatematicaUniversità di Roma ‘La Sapienza’RomeItaly

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