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


SIGMA 8 (2012), 066, 29 pages      arXiv:1210.0651      https://doi.org/10.3842/SIGMA.2012.066
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

A New Class of Solvable Many-Body Problems

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

Received June 27, 2012, in final form September 20, 2012; Published online October 02, 2012

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×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.

Key words: integrable dynamical systems; solvable dynamical systems; solvable Newtonian many-body problems; integrable Newtonian many-body problems; isochronous dynamical systems.

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References

  1. 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.
  2. Olshanetsky M.A., Perelomov A.M., Completely integrable Hamiltonian systems connected with semisimple Lie algebras, Invent. Math. 37 (1976), 93-108.
  3. Olshanetsky M.A., Perelomov A.M., Explicit solutions of some completely integrable systems, Lett. Nuovo Cimento 17 (1976), 97-101.
  4. Olshanetsky M.A., Perelomov A.M., Classical integrable finite-dimensional systems related to Lie algebras, Phys. Rep. 71 (1981), 313-400.
  5. Calogero F., Classical many-body problems amenable to exact treatments, Lecture Notes in Physics. New Series m: Monographs, Vol. 66, Springer-Verlag, Berlin, 2001.
  6. Calogero F., Isochronous systems, Oxford University Press, Oxford, 2008.
  7. Bruschi M., Calogero F., Droghei R., Integrability, analyticity, isochrony, equilibria, small oscillations, and Diophantine relations: results from the stationary Burgers hierarchy, J. Phys. A: Math. Theor. 42 (2009), 475202, 9 pages.
  8. Bruschi M., Calogero F., Droghei R., Integrability, analyticity, isochrony, equilibria, small oscillations, and Diophantine relations: results from the stationary Korteweg-de Vries hierarchy, J. Math. Phys. 50 (2009), 122701, 19 pages.
  9. Calogero F., A new class of solvable dynamical systems, J. Math. Phys. 49 (2008), 052701, 9 pages.
  10. Calogero F., Two new solvable dynamical systems of goldfish type, J. Nonlinear Math. Phys. 17 (2010), 397-414.
  11. Calogero F., Isochronous dynamical systems, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 369 (2011), 1118-1136.
  12. Calogero F., A new goldfish model, Theoret. and Math. Phys. 167 (2011), 714-724.
  13. Calogero F., Another new goldfish model, Theoret. and Math. Phys. 171 (2012), 629-640.
  14. Calogero F., An integrable many-body problem, J. Math. Phys. 52 (2011), 102702, 5 pages.
  15. Calogero F., New solvable many-body model of goldfish type, J. Nonlinear Math. Phys. 19 (2012), 1250006, 19 pages.
  16. Calogero F., Two quite similar matrix ODEs and the many-body problems related to them, Int. J. Geom. Methods Mod. Phys. 9 (2012), 1260002, 6 pages.
  17. Calogero F., Another new solvable many-body model of goldfish type, SIGMA 8 (2012), 046, 17 pages, arXiv:1207.4850.
  18. Calogero F., Iona S., Isochronous dynamical system and Diophantine relations. I, J. Nonlinear Math. Phys. 16 (2009), 105-116.
  19. Calogero F., Leyvraz F., Examples of isochronous Hamiltonians: classical and quantal treatments, J. Phys. A: Math. Theor. 41 (2008), 175202, 11 pages.
  20. Calogero F., Leyvraz F., A new class of isochronous dynamical systems, J. Phys. A: Math. Theor. 41 (2008), 295101, 14 pages.
  21. Calogero F., Leyvraz F., Isochronous oscillators, J. Nonlinear Math. Phys. 17 (2010), 103-110.
  22. Calogero F., Leyvraz F., Solvable systems of isochronous, multi-periodic or asymptotically isochronous nonlinear oscillators, J. Nonlinear Math. Phys. 17 (2010), 111-120.
  23. Droghei R., Calogero F., Ragnisco O., An isochronous variant of the Ruijsenaars-Toda model: equilibrium configurations, behavior in their neighborhood, Diophantine relations, J. Phys. A: Math. Theor. 42 (2009), 445207, 9 pages.
  24. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G. (Editors), Higher transcendental functions, Vol. I, Bateman Manuscript Project, McGraw-Hill Book Co., New York, 1953.

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