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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.
MSC: 37J35;
37C27;
70F10;
70H08 Received: May 3, 2012; in final form
July 17, 2012; Published online
July 20, 2012
Language: English
DOI:
10.3842/SIGMA.2012.046
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