
SIGMA 8 (2012), 066, 29 pages arXiv:1210.0651
http://dx.doi.org/10.3842/SIGMA.2012.066
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”
A New Class of Solvable ManyBody 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 Nbody problems is identified.
They describe N unitmass point particles whose timeevolution, generally
taking place in the complex plane, is characterized by Newtonian equations of motion ''of goldfish type'' (acceleration equal
force, with specific velocitydependent onebody and twobody forces)
featuring several arbitrary coupling constants. The corresponding
initialvalue problems are solved by finding the eigenvalues of a
timedependent 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 manybody problems; integrable Newtonian manybody
problems; isochronous dynamical systems.
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