New many-body problems in the plane with periodic solutions
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2004
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IOP Publishing
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Abstract
In this paper we discuss a family of toy models for many-body interactions including velocity-dependent forces. By generalizing a construction due to Calogero, we obtain a class of N-body problems in the plane which have periodic orbits for a large class of initial conditions. The two- and three-body cases (N = 2, 3) are exactly solvable, with all solutions being periodic, and we present their explicit solutions. For N greater than or equal to 4 Painleve analysis indicates that the system should not be integrable, and some periodic and non-periodic trajectories are calculated numerically. The construction can be generalized to a broad class of systems, and the mechanism which describes the transition to orbits with higher periods, and eventually to aperiodic or even chaotic orbits, could be present in more realistic models with a mixed phase space. This scenario is different from the onset of chaos by a sequence of Hopf bifurcations.
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© IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.
This work was begun during the scientific gathering on Integrable Systems organized at the Centro Internacional de Ciencias, Cuernavaca in November–December 2002. The authors thank the staff of the CiC, especially the director Thomas Seligman, for providing the pleasant atmosphere in which the work was initiated. The research of DGU is partially supported by the CRM-ISM grant and the Spanish Ministry of Education.ANWH thanks the London Mathematical Society for supporting his stay in Mexico with a Scheme 5 grant. The authors thank Francesco Calogero for teaching them many of the developments contained in this work.