2014
DOI: 10.1007/s10884-014-9401-2
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Platonic Polyhedra, Periodic Orbits and Chaotic Motions in the $$N$$ N -body Problem with Non-Newtonian Forces

Abstract: We consider the N -body problem with interaction potential U α = 1 |x i −x j | α for α > 1. We assume that the particles have all the same mass and that N is the order |R| of the rotation group R of one of the five Platonic polyhedra. We study motions that, up to a relabeling of the N particles, are invariant under R. By variational techniques we prove the existence of periodic and chaotic motions.

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Cited by 4 publications
(6 citation statements)
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“…be the angular momentum ofq with respect to the z axis. As observed in Remark 1, it is constant in (c, d), because thereinq solves (12). By the conservation of the energy, we havē…”
Section: 2mentioning
confidence: 77%
See 1 more Smart Citation
“…be the angular momentum ofq with respect to the z axis. As observed in Remark 1, it is constant in (c, d), because thereinq solves (12). By the conservation of the energy, we havē…”
Section: 2mentioning
confidence: 77%
“…On the other hand, when looking for new selected trajectories in Celestial Machanics, one often seeks minimizers of the action in some set of functions sharing a prescribed topological behaviour, as, for example, in [6,7,12,13,14,20,24,25]. In such a situations, Marchal's lemma cannot be employed because the average argument may destroy the topological constraint, and this, usually, makes impossible to deduce any conclusive information.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…where u I : [0, T ] → R 3 is the motion of an arbitrarily selected satellite, that we call the generating particle 1 . In the following we shall discuss some examples, considering the Z 4 group (leading to the Hip-Hop solution [1,10] with a central body), the Klein group Z 2 × Z 2 and the symmetry groups of Platonic polyhedra [18,21,22]. If we restrict the action A α to G , then it depends only on the motion of the generating particle:…”
Section: } and Define The Lagrangian Action Functional Asmentioning
confidence: 99%
“…We shall show some examples, with different symmetry constraints. In particular, we shall consider symmetries defined by the group Z 4 (leading to the Hip-Hop solution [10]), by Z 2 × Z 2 , and by the rotation groups of Platonic polyhedra (used for instance in [18,[20][21][22]24]). -convergence was already applied to the N -body problem in [21], where the authors considered the exponent α of the potential as a parameter, and studied the behavior of the minimizers as α → +∞.…”
Section: Introductionmentioning
confidence: 99%
“…We shall show some examples, with different symmetry constraints. In particular, we shall consider symmetries defined by the group Z 4 (leading to the Hip-Hop solution [10]), by Z 2 × Z 2 , and by the rotation groups of Platonic polyhedra (used for instance in [18,[20][21][22]24]). Γ-convergence was already applied to the N -body problem in [21], where the authors considered the exponent α of the potential as a parameter, and studied the behavior of the minimizers as α → +∞.…”
Section: Introductionmentioning
confidence: 99%