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Эта публикация цитируется в 2 научных статьях (всего в 2 статьях)
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
Аннотация:
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.
Ключевые слова:
nonlinear discrete-time dynamical systems, integrable and solvable maps, isochronous discrete-time dynamical systems, discrete-time dynamical systems of goldfish type.
Поступила: 3 мая 2012 г.; в окончательном варианте 17 июля 2012 г.; опубликована 20 июля 2012 г.
Образец цитирования:
Francesco Calogero, “Another new solvable many-body model of goldfish type”, SIGMA, 8 (2012), 046, 17 pp.
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/sigma723 https://www.mathnet.ru/rus/sigma/v8/p46
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