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Three New Classes of Solvable N-Body Problems of Goldfish Type with Many Arbitrary Coupling Constants

1
Physics Department, University of Rome “La Sapienza”, 00185 Rome, Italy
2
Istituto Nazionale di Fisica Nucleare, Sezione di Roma, 00185 Rome, Italy
Academic Editor: Charles F. Dunkl
Symmetry 2016, 8(7), 53; https://doi.org/10.3390/sym8070053
Received: 31 March 2016 / Revised: 16 June 2016 / Accepted: 17 June 2016 / Published: 24 June 2016
(This article belongs to the Special Issue Symmetry in Orthogonal Polynomials)
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Abstract

Three new classes of N-body problems of goldfish type are identified, with N an arbitrary positive integer ( N 2 ). These models are characterized by nonlinear Newtonian (“accelerations equal forces”) equations of motion describing N equal point-particles moving in the complex z-plane. These highly nonlinear equations feature many arbitrary coupling constants, yet they can be solved by algebraic operations. Some of these N-body problems are isochronous, their generic solutions being all completely periodic with an overall period T independent of the initial data (but quite a few of these solutions are actually periodic with smaller periods T / p with p a positive integer); other models are isochronous for an open region of initial data, while the motions for other initial data are not periodic, featuring instead scattering phenomena with some of the particles incoming from, or escaping to, infinity in the remote past or future. View Full-Text
Keywords: solvable many-body problems; integrable many-body problems; integrable dynamical systems; solvable many-body problems of goldfish type solvable many-body problems; integrable many-body problems; integrable dynamical systems; solvable many-body problems of goldfish type
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Calogero, F. Three New Classes of Solvable N-Body Problems of Goldfish Type with Many Arbitrary Coupling Constants. Symmetry 2016, 8, 53.

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