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 SIGMA, 2012, Volume 8, 046, 17 pp. (Mi sigma723)

Another new solvable many-body model of goldfish type

Francesco Calogeroab

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 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 $t\to\infty$ (“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 discrete-time dynamical systems, integrable and solvable maps, isochronous discrete-time dynamical systems, discrete-time dynamical systems of goldfish type.

DOI: https://doi.org/10.3842/SIGMA.2012.046

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ArXiv: 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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Citation: Francesco Calogero, “Another new solvable many-body model of goldfish type”, SIGMA, 8 (2012), 046, 17 pp.

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\Bibitem{Cal12} \by Francesco Calogero \paper Another new solvable many-body model of goldfish type \jour SIGMA \yr 2012 \vol 8 \papernumber 046 \totalpages 17 \mathnet{http://mi.mathnet.ru/sigma723} \crossref{https://doi.org/10.3842/SIGMA.2012.046} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2958984} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000306612500001} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84864837807} 

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This publication is cited in the following articles:
1. Francesco Calogero, Ge Yi, “A new class of solvable many-body problems”, SIGMA, 8 (2012), 066, 29 pp.
2. 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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