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SIGMA 8 (2012), 046, 17 pages arXiv:1207.4850
https://doi.org/10.3842/SIGMA.2012.046
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”
Another New Solvable Many-Body 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 many-body problem is identified.
It is characterized by nonlinear Newtonian equations of motion
(''acceleration equal force'') featuring one-body and two-body
velocity-dependent forces ''of goldfish type'' which determine the motion of
an arbitrary number $N$ of unit-mass point-particles 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 time-dependent $N\times N$ 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 discrete-time dynamical systems; integrable
and solvable maps; isochronous discrete-time dynamical systems;
discrete-time dynamical systems of goldfish type.
pdf (372 kb)
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