Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 046, 17 pages      arXiv:1207.4850      http://dx.doi.org/10.3842/SIGMA.2012.046
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”

Another New Solvable Many-Body Model of Goldfish Type

Francesco Calogero
Physics Department, University of Rome ''La Sapienza'', Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy

Received May 03, 2012, in final form July 17, 2012; Published online July 20, 2012

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\rightarrow \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.

Key words: nonlinear discrete-time dynamical systems; integrable and solvable maps; isochronous discrete-time dynamical systems; discrete-time dynamical systems of goldfish type.

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References

  1. Calogero F., Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419-436, Erratum, J. Math. Phys. 37 (1996), 3646.
  2. Moser J., Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16 (1975), 197-220.
  3. Calogero F., A solvable N-body problem in the plane. I, J. Math. Phys. 37 (1996), 1735-1759.
  4. Calogero F., Françoise J.P., Hamiltonian character of the motion of the zeros of a polynomial whose coefficients oscillate over time, J. Phys. A: Math. Gen. 30 (1997), 211-218.
  5. Calogero F., Isochronous systems, Oxford University Press, Oxford, 2008.
  6. Calogero F., Motion of poles and zeros of special solutions of nonlinear and linear partial differential equations and related "solvable" many-body problems, Nuovo Cimento B 43 (1978), 177-241.
  7. Calogero F., The neatest many-body problem amenable to exact treatments (a "goldfish"?), Phys. D 152/153 (2001), 78-84.
  8. Olshanetsky M.A., Perelomov A.M., Explicit solution of the Calogero model in the classical case and geodesic flows on symmetric spaces of zero curvature, Lett. Nuovo Cimento 16 (1976), 333-339.
  9. Olshanetsky M.A., Perelomov A.M., Classical integrable finite-dimensional systems related to Lie algebras, Phys. Rep. 71 (1981), 313-400.
  10. Perelomov A.M., Integrable systems of classical mechanics and Lie algebras, Birkhäuser Verlag, Basel, 1990.
  11. Calogero F., Classical many-body problems amenable to exact treatments, Lecture Notes in Physics. New Series m: Monographs, Vol. 66, Springer-Verlag, Berlin, 2001.
  12. Calogero F., Two new solvable dynamical systems of goldfish type, J. Nonlinear Math. Phys. 17 (2010), 397-414.
  13. Calogero F., A new goldfish model, Theoret. and Math. Phys. 167 (2011), 714-724.
  14. Calogero F., Another new goldfish model, Theoret. and Math. Phys. 171 (2012), 629-640.
  15. Calogero F., New solvable many-body model of goldfish type, J. Nonlinear Math. Phys. 19 (2012), 1250006, 19 pages.
  16. Gómez-Ullate D., Sommacal M., Periods of the goldfish many-body problem, J. Nonlinear Math. Phys. 12 (2005), suppl. 1, 351-362.

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