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On the existence of collisionless equivariant minimizers for the classical n-body problem

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Abstract

We show that the minimization of the Lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n-body problem, provided that some simple conditions on the symmetry group are satisfied. More precisely, we give a fairly general condition on symmetry groups G of the loop space Λ for the n-body problem (with potential of homogeneous degree -α, with α>0) which ensures that the restriction of the Lagrangian action \(\mathcal{A}\) to the space ΛG of G-equivariant loops is coercive and its minimizers are collisionless, without any strong force assumption. In proving that local minima of ΛG are free of collisions we develop an averaging technique based on Marchal’s idea of replacing some of the point masses with suitable shapes (see [10]). As an application, several new orbits can be found with some appropriate choice of G. Furthermore, the result can be used to give a simplified and unitary proof of the existence of many already known minimizing periodic orbits.

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Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133, Milano, Italy

    Davide L. Ferrario

  2. Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via Bicocca degli Arcimboldi, 8, 20126, Milano, Italy

    Susanna Terracini

Authors
  1. Davide L. Ferrario
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  2. Susanna Terracini
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Correspondence to Davide L. Ferrario or Susanna Terracini.

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Mathematics Subject Classification (2000)

70F10, 70F16, 37C80, 70G75

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Ferrario, D., Terracini, S. On the existence of collisionless equivariant minimizers for the classical n-body problem. Invent. math. 155, 305–362 (2004). https://doi.org/10.1007/s00222-003-0322-7

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