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 SIGMA, 2012, Volume 8, 066, 29 pp. (Mi sigma743)

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.

DOI: https://doi.org/10.3842/SIGMA.2012.066

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ArXiv: 1210.0651
MSC: 70F10; 70H06; 37J35; 37K10
Received: June 27, 2012; in final form September 20, 2012; Published online October 2, 2012
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Citation: Francesco Calogero, Ge Yi, “A new class of solvable many-body problems”, SIGMA, 8 (2012), 066, 29 pp.

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Oksana Bihun, Francesco Calogero, “Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces”, SIGMA, 9 (2013), 059, 18 pp.
2. Bihun O. Calogero F., “Equilibria of a Solvable N-Body Problem and Related Properties of the N Numbers X(N) at Which the Jacobi Polynomial of Order N Has the Same Value”, J. Nonlinear Math. Phys., 20:4 (2013), 539–551
3. 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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