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

SIGMA 8 (2012), 066, 29 pages      arXiv:1210.0651      https://doi.org/10.3842/SIGMA.2012.066
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

A New Class of Solvable Many-Body Problems

Francesco Calogero and Ge Yi
Physics Department, University of Rome ''La Sapienza'', Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy

Received June 27, 2012, in final form September 20, 2012; Published online October 02, 2012

Abstract
A new class of solvable N-body problems is identified. They describe N unit-mass point particles whose time-evolution, generally taking place in the complex plane, is characterized by Newtonian equations of motion ''of goldfish type'' (acceleration equal force, with specific velocity-dependent one-body and two-body forces) featuring several arbitrary coupling constants. The corresponding initial-value problems are solved by finding the eigenvalues of a time-dependent N×N matrix U(t) explicitly defined in terms of the initial positions and velocities of the N particles. Some of these models are asymptotically isochronous, i.e. in the remote future they become completely periodic with a period T independent of the initial data (up to exponentially vanishing corrections). Alternative formulations of these models, obtained by changing the dependent variables from the N zeros of a monic polynomial of degree N to its N coefficients, are also exhibited.

Key words: integrable dynamical systems; solvable dynamical systems; solvable Newtonian many-body problems; integrable Newtonian many-body problems; isochronous dynamical systems.

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