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3 papers
A new class of solvable many-body problems
Francesco Calogeroab,
Ge Yiab a Physics Department, University of Rome "La Sapienza", Italy
b Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy
Abstract:
A
new class of
solvable $N$-body problems is identified. They describe
$N$ unit-mass point particles whose time-evolution, generally taking place in the
complex plane, is characterized by
Newtonian equations of motion “of goldfish type” (acceleration equal force, with specific velocity-dependent one-body and two-body forces) featuring several arbitrary coupling constants. The corresponding initial-value problems are solved by finding the eigenvalues of a time-dependent
$N\times N$ matrix
$U(t)$ explicitly defined in terms of the initial positions and velocities of the
$N$ particles. Some of these models are
asymptotically isochronous, i.e. in the remote future they become completely periodic with a period
$T$ independent of the initial data (up to exponentially vanishing corrections). Alternative formulations of these models, obtained by changing the dependent variables from the
$N$ zeros of a monic polynomial of degree
$N$ to its
$N$ coefficients, are also exhibited.
Keywords:
integrable dynamical systems; solvable dynamical systems; solvable Newtonian many-body problems; integrable Newtonian many-body problems; isochronous dynamical systems.
MSC: 70F10;
70H06;
37J35;
37K10 Received: June 27, 2012; in final form
September 20, 2012; Published online
October 2, 2012
Language: English
DOI:
10.3842/SIGMA.2012.066
Fulltext:
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