
SIGMA, 2012, Volume 8, 046, 17 pages
(Mi sigma723)




This article is cited in 2 scientific papers (total in 2 papers)
Another new solvable manybody model of goldfish type
Francesco Calogero^{ab} ^{a} Physics Department, University of Rome ''La Sapienza'', Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy
^{b} INFN — National Institute of Nuclear Physics
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\to\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.
Keywords:
nonlinear discretetime dynamical systems, integrable and solvable maps, isochronous discretetime dynamical systems, discretetime dynamical systems of goldfish type.
DOI:
https://doi.org/10.3842/SIGMA.2012.046
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1207.4850
MSC: 37J35; 37C27; 70F10; 70H08 Received: May 3, 2012; in final form July 17, 2012; Published online July 20, 2012
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Francesco Calogero, “Another new solvable manybody model of goldfish type”, SIGMA, 8 (2012), 046, 17 pp.
Citation in format AMSBIB
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This publication is cited in the following articles:

Francesco Calogero, Ge Yi, “A new class of solvable manybody problems”, SIGMA, 8 (2012), 066, 29 pp.

Calogero F., Yi G., “Polynomials Satisfying Functional and Differential Equations and Diophantine Properties of their Zeros”, Lett. Math. Phys., 103:6 (2013), 629–651

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