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Platonic Polyhedra, Periodic Orbits and Chaotic Motions in the \(N\)-body Problem with Non-Newtonian Forces

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Abstract

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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Notes

  1. \(\mathcal {T}\) the rotation group of the Tetrahedron of order \(12\), \(\mathcal {O}\) the rotation group of Hexahedron and Octahedron of order \(24\) and \(\mathcal {I}\) the group of Icosahedron and Dodecahedron of order \(60\).

  2. Animations of (\(\mathcal {T}, \nu _4\)) (Electronic supplementary material 3), (\(\mathcal {O}, \nu _7\)) (Electronic supplementary material 6), (\(\mathcal {I}, \nu _3\)) (Electronic supplementary material 2) are available in the online supplementary material.

  3. For animation of the motions in Table 2 see [3] and the online supplementary material: (\(\mathcal {O}, \nu _1\)) (Electronic supplementary material 4), (\(\mathcal {O}, \nu _2\)) (Electronic supplementary material 5), (\(\mathcal {I}, \nu _1\)) (Electronic supplementary material 1).

  4. \(r_p\) is the radius of the circumcircle of a regular polygon with \(o(p)\) sides of unitary length.

  5. See also the online supplementary materials (\(\mathcal {O}, \nu _1\)) (Electronic supplementary material 7), (\(\mathcal {O}, \nu _2\)) (Electronic supplementary material 8).

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Authors and Affiliations

  1. Dip. di Matematica Pura ed Applicata, Università di L’Aquila, L’Aquila, Italy

    Giorgio Fusco

  2. Dip. di Matematica, Università di Pisa, Pisa, Italy

    Giovanni F. Gronchi

Authors
  1. Giorgio Fusco
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  2. Giovanni F. Gronchi
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Correspondence to Giorgio Fusco.

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Fusco, G., Gronchi, G.F. Platonic Polyhedra, Periodic Orbits and Chaotic Motions in the \(N\)-body Problem with Non-Newtonian Forces. J Dyn Diff Equat 26, 817–841 (2014). https://doi.org/10.1007/s10884-014-9401-2

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  • DOI: https://doi.org/10.1007/s10884-014-9401-2

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