
SIGMA 8 (2012), 046, 17 pages arXiv:1207.4850
http://dx.doi.org/10.3842/SIGMA.2012.046
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”
Another New Solvable ManyBody Model of Goldfish Type
Francesco Calogero
Physics Department, University of Rome ''La Sapienza'', Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy
Received May 03, 2012, in final form July 17, 2012; Published online July 20, 2012
Abstract
A new solvable manybody problem is identified.
It is characterized by nonlinear Newtonian equations of motion
(''acceleration equal force'') featuring onebody and twobody
velocitydependent forces ''of goldfish type'' which determine the motion of
an arbitrary number $N$ of unitmass pointparticles in a plane. The $N$
(generally complex) values $z_{n}( t) $ at time $t$ of
the $N$ coordinates of these moving particles are given by the $N$
eigenvalues of a timedependent $N\times N$ matrix $U( t) $
explicitly known in terms of the $2N$ initial data $z_{n}( 0) $
and $\dot{z}_{n}(0) $. This model comes in two different
variants, one featuring 3 arbitrary coupling constants, the other only 2;
for special values of these parameters all solutions are completely
periodic with the same period independent of the initial data (''isochrony''); for other special values of these parameters this property
holds up to corrections vanishing exponentially as $t\rightarrow \infty $ (''asymptotic isochrony''). Other isochronous variants of
these models are also reported. Alternative formulations, obtained by
changing the dependent variables from the $N$ zeros of a monic polynomial of
degree $N$ to its $N$ coefficients, are also exhibited. Some mathematical
findings implied by some of these results – such as Diophantine
properties of the zeros of certain polynomials – are outlined, but their
analysis is postponed to a separate paper.
Key words:
nonlinear discretetime dynamical systems; integrable
and solvable maps; isochronous discretetime dynamical systems;
discretetime dynamical systems of goldfish type.
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