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September 2012, 32(9): 3245-3301. doi: 10.3934/dcds.2012.32.3245
Symbolic dynamics for the $N$-centre problem at negative energies
Nicola Soave 1, and Susanna Terracini 2,
Università di Milano Bicocca - Dipartimento di Matematica e Applicazioni, Via Cozzi 53, 20125 Milano, Italy
Università di Milano Bicocca, Dipartimento di Matematica e Applicazioni, Via Cozzi 53, 20125 Milano
Received February 2012 Revised March 2012 Published April 2012
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.
Keywords: chaotic motions, $N$-centre problem, symbolic dynamics, Levi-Civita regularization..
Mathematics Subject Classification: Primary: 70F10, 37N05; Secondary: 70F15, 37J3.
Citation: Nicola Soave, Susanna Terracini. Symbolic dynamics for the $N$-centre problem at negative energies. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3245-3301. doi: 10.3934/dcds.2012.32.3245
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Nicola Soave Susanna Terracini