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Regular and Chaotic Dynamics

, Volume 21, Issue 5, pp 522–530 | Cite as

Computing hyperbolic choreographies

  • Hadrien Montanelli
Article

Abstract

An algorithm is presented for numerical computation of choreographies in spaces of constant negative curvature in a hyperbolic cotangent potential, extending the ideas given in a companion paper [14] for computing choreographies in the plane in a Newtonian potential and on a sphere in a cotangent potential. Following an idea of Diacu, Pérez-Chavela and Reyes Victoria [9], we apply stereographic projection and study the problem in the Poincaré disk. Using approximation by trigonometric polynomials and optimization methods with exact gradient and exact Hessian matrix, we find new choreographies, hyperbolic analogues of the ones presented in [14]. The algorithm proceeds in two phases: first BFGS quasi-Newton iteration to get close to a solution, then Newton iteration for high accuracy.

Keywords

choreographies curved n-body problem trigonometric interpolation quasi-Newton methods Newton’s method 

MSC2010 numbers

70F10 70F15 70H12 

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Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Oxford University Mathematical InstituteOxfordUK

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