Symbolic dynamics for the $N$-centre problem at negative energies

Nicola Soave; Susanna Terracini

doi:10.3934/dcds.2012.32.3245

Article Contents

2012, Volume 32, Issue 9: 3245-3301. Doi: 10.3934/dcds.2012.32.3245

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Symbolic dynamics for the $N$-centre problem at negative energies

Nicola Soave^1, and
Susanna Terracini^2,

1.
Università di Milano Bicocca - Dipartimento di Matematica e Applicazioni, Via Cozzi 53, 20125 Milano
2.
Università di Milano Bicocca, Dipartimento di Matematica e Applicazioni, Via Cozzi 53, 20125 Milano

Received: January 31, 2012

Revised: February 29, 2012

Published: August 2012

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Abstract

We consider the planar $N$-centre problem, with homogeneous potentials of degree $-\alpha < 0$, $\alpha \in [1,2)$. We prove the existence of infinitely many collisions-free periodic solutions with negative and small energy, for any distribution of the centres inside a compact set. The proof is based upon topological, variational and geometric arguments. The existence result allows to characterize the associated dynamical system with a symbolic dynamics, where the symbols are the partitions of the $N$ centres in two non-empty sets.

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Mathematics Subject Classification: Primary: 70F10, 37N05; Secondary: 70F15, 37J30.

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