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


SIGMA 7 (2011), 082, 35 pages      arXiv:1108.4492      https://doi.org/10.3842/SIGMA.2011.082
Contribution to the Proceedings of the Conference “Symmetries and Integrability of Difference Equations (SIDE-9)”

Discrete-Time Goldfishing

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

Received May 04, 2011, in final form July 29, 2011; Published online August 23, 2011

Abstract
The original continuous-time ''goldfish'' dynamical system is characterized by two neat formulas, the first of which provides the N Newtonian equations of motion of this dynamical system, while the second provides the solution of the corresponding initial-value problem. Several other, more general, solvable dynamical systems ''of goldfish type'' have been identified over time, featuring, in the right-hand (''forces'') side of their Newtonian equations of motion, in addition to other contributions, a velocity-dependent term such as that appearing in the right-hand side of the first formula mentioned above. The solvable character of these models allows detailed analyses of their behavior, which in some cases is quite remarkable (for instance isochronous or asymptotically isochronous). In this paper we introduce and discuss various discrete-time dynamical systems, which are as well solvable, which also display interesting behaviors (including isochrony and asymptotic isochrony) and which reduce to dynamical systems of goldfish type in the limit when the discrete-time independent variable l=0,1,2,... becomes the standard continuous-time independent variable t, 0≤t<∞.

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., 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.
  2. Calogero F., The neatest many-body problem amenable to exact treatments (a "goldfish"?), Phys. D 152/153 (2001), 78-84.
  3. Ruijsenaars S.N.M., Schneider H., A new class of integrable systems and its relation to solitons, Ann. Physics 170 (1986), 370-405.
  4. Calogero F., Classical many-body problems amenable to exact treatments, Lecture Notes in Physics, New Series m: Monographs, Vol. 66, Springer-Verlag, Berlin, 2001.
  5. Gómez-Ullate D., Sommacal M., Periods of the goldfish many-body problem, J. Nonlinear Math. Phys. 12 (2005), suppl. 1, 351-362.
  6. Calogero F., Isochronous systems, Oxford University Press, Oxford, 2008.
  7. Calogero F., Two new solvable dynamical systems of goldfish type, J. Nonlinear Math. Phys. 17 (2010), 397-414.
  8. Calogero F., A new goldfish model, Theoret. and Math. Phys. 167 (2011), 714-724.
  9. Calogero F., Another new goldfish model, Theoret. and Math. Phys., to appear.
  10. Calogero F., The discrete-time goldfish, unpublished.
  11. Veselov A.P., Integrable maps, Russian Math. Surveys 46 (1991), no. 5, 1-51.
  12. Clarkson P.A., Nijhoff F.W. (Editors), Symmetries and integrability of difference equations, London Mathematical Society Lecture Notes Series, Vol. 255, Cambridge University Press, Cambridge, 1999.
  13. Suris Yu.B., The problem of integrable discretization: Hamiltonian approach, Progress in Mathematics, Vol. 219, Birkhäuser, Basel, 2003.
  14. Ragnisco O., Discrete integrable systems, in Encyclopedia of Mathematical Physics, Vol. 3, Elsevier, Oxford, 2006, 59-65.
  15. Bobenko A.I., Suris Yu.B., Discrete differential geometry. Integrable structure, Graduate Studies in Mathematics, Vol. 98, American Mathematical Society, Providence, RI, 2008.
  16. Nijhoff F.W., Ragnisco O., Kuznetsov V.B., Integrable time-discretization of the Ruijsenaars-Schneider model, Comm. Math. Phys. 176 (1996), 681-700, hep-th/9412170.
  17. Nijhoff F.W., Pang G.D., A time-discretized version of the Calogero-Moser model, Phys. Lett. A 191 (1994), 101-107, hep-th/9403052.
    Nijhoff F.W., Pang G.D., Discrete-time Calogero-Moser model and lattice KP equations, in Symmetries and Integrability of Difference Equations (Estérel, PQ, 1994), Editors D. Levi, L. Vinet and P. Winternitz, CRM Proc. Lecture Notes, Vol. 9, Amer. Math. Soc., Providence, RI, 1996, 253-264, hep-th/9409071.
  18. Suris Yu.B., Time discretization of F. Calogero's "goldfish" system, J. Nonlinear Math. Phys. 12 (2005), suppl. 1, 633-647.
  19. 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.

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