Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 059, 18 pages      arXiv:1310.2335      https://doi.org/10.3842/SIGMA.2013.059

Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces

Oksana Bihun a and Francesco Calogero b
a) Department of Mathematics, Concordia College at Moorhead, MN, USA
b) Physics Department, University of Rome ''La Sapienza'', Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy

Received June 07, 2013, in final form October 02, 2013; Published online October 09, 2013

Abstract
The class of solvable many-body problems ''of goldfish type'' is extended by including (the additional presence of) three-body forces. The solvable N-body problems thereby identified are characterized by Newtonian equations of motion featuring 19 arbitrary ''coupling constants''. Restrictions on these constants are identified which cause these systems – or appropriate variants of them – to be isochronous or asymptotically isochronous, i.e. all their solutions to be periodic with a fixed period (independent of the initial data) or to have this property up to contributions vanishing exponentially as t→ ∞.

Key words: many-body problems; N-body problems; partial differential equations; isochronous systems.

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