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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