
SIGMA, 2012, Volume 8, 066, 29 pages
(Mi sigma743)




This article is cited in 3 scientific papers (total in 3 papers)
A new class of solvable manybody problems
Francesco Calogero^{ab}, Ge Yi^{ab} ^{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 $N$ unitmass point particles whose timeevolution, generally taking place in the complex plane, is characterized by Newtonian equations of motion “of goldfish type” (acceleration equal force, with specific velocitydependent onebody and twobody forces) featuring several arbitrary coupling constants. The corresponding initialvalue problems are solved by finding the eigenvalues of a timedependent $N\times N$ 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 manybody problems; integrable Newtonian manybody 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 manybody problems”, SIGMA, 8 (2012), 066, 29 pp.
Citation in format AMSBIB
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\by Francesco Calogero, Ge Yi
\paper A new class of solvable manybody problems
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\papernumber 066
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This publication is cited in the following articles:

Oksana Bihun, Francesco Calogero, “Solvable ManyBody Models of Goldfish Type with One, Two and ThreeBody Forces”, SIGMA, 9 (2013), 059, 18 pp.

Bihun O. Calogero F., “Equilibria of a Solvable NBody 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

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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