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Equivariant optimisation for the gravitational $n$-body problem: a computational factory of symmetric orbits

Vivina Barutello, Mattia G. Bergomi, Gian Marco Canneori, Roberto Ciccarelli, Davide L. Ferrario, Susanna Terracini, Pietro Vertechi

Abstract

In this paper we present \texttt{SymOrb.jl}, a software which combines group representation theory and variational methods to provide numerical solutions of singular dynamical systems of paramount relevance in Celestial Mechanics and other interacting particles models. Among all, it prepares for large-scale search of symmetric periodic orbits for the classical $n$-body problem and their classification, paving the way towards a computational validation of Poincaré conjecture about the density of periodic orbits. Through the accessible language of Julia, \texttt{SymOrb.jl} offers a unified implementation of an earlier version. This paper provides theoretical and practical guidelines for the specific approach we adopt, complemented with examples.

Equivariant optimisation for the gravitational $n$-body problem: a computational factory of symmetric orbits

Abstract

In this paper we present \texttt{SymOrb.jl}, a software which combines group representation theory and variational methods to provide numerical solutions of singular dynamical systems of paramount relevance in Celestial Mechanics and other interacting particles models. Among all, it prepares for large-scale search of symmetric periodic orbits for the classical -body problem and their classification, paving the way towards a computational validation of Poincaré conjecture about the density of periodic orbits. Through the accessible language of Julia, \texttt{SymOrb.jl} offers a unified implementation of an earlier version. This paper provides theoretical and practical guidelines for the specific approach we adopt, complemented with examples.

Paper Structure

This paper contains 24 sections, 12 theorems, 88 equations, 1 figure.

Table of Contents

  1. Conclusions and perspectives
  2. Notations
  3. Symmetries as group actions
  4. Action of finite groups on $\Lambda$
  5. Equivariant variational principles
  6. Reducible, bound to collisions and admissible actions on $\Lambda$
  7. Classification of the group action on $\Lambda$ and fundamental domain
  8. $\boldsymbol G$-equivariant critical points of the action functional
  9. Approximating critical points of $\mathcal{A}$
  10. Configuration space and projectors
  11. The action functional
  12. The kinetic part
  13. The potential part
  14. Numerical algorithm
  15. Extension to the whole period
  16. ...and 9 more sections

Key Result

Proposition 2.3

Let $G$ be a finite group, with orthogonal representations $\rho\colon G\to O(d)$, $\tau\colon G\to O(2)$ and $\sigma\colon G\to\Sigma_n$ satisfying property:sigma. Then the following assertions hold true:

Figures (1)

  • Figure 1: The visualisation periodic orbit found by the optimisation routine described in section \ref{['sec:example_session']}.

Theorems & Definitions (44)

  • Remark 1
  • Definition 1.1: Group action
  • Definition 1.2: Isotropy groups and projectors
  • Remark 1.3
  • Example 1.4: $X$ finite, action of $D_{2n}$
  • Example 1.5: $X$ linear space
  • Remark 2.1
  • Example 2.2: $G=D_6$
  • Proposition 2.3
  • Lemma 2.4
  • ...and 34 more