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


SIGMA 4 (2008), 031, 18 pages      arXiv:0803.1824      https://doi.org/10.3842/SIGMA.2008.031
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Recent Applications of the Theory of Lie Systems in Ermakov Systems

José F. Cariñena, Javier de Lucas and Manuel F. Rañada
Department of Theoretical Physics, University of Zaragoza, 50.009 Zaragoza, Spain

Received November 02, 2007, in final form February 04, 2008; Published online March 12, 2008

Abstract
We review some recent results of the theory of Lie systems in order to apply such results to study Ermakov systems. The fundamental properties of Ermakov systems, i.e. their superposition rules, the Lewis-Ermakov invariants, etc., are found from this new perspective. We also obtain new results, such as a new superposition rule for the Pinney equation in terms of three solutions of a related Riccati equation.

Key words: superposition rule; Pinney equation; Ermakov systems.

pdf (276 kb)   ps (181 kb)   tex (21 kb)

References

  1. Ermakov V.P., Second-order differential equations. Conditions of complete integrability, Univ. Isz. Kiev Series III 9 (1880), 1-25 (translation by A.O. Harin).
  2. Lie S., Vorlesungen über continuierliche Gruppen mit Geometrischen und anderen Anwendungen, edited and revised by G. Scheffers, Teubner, Leipzig, 1893.
  3. Ibragimov N.H., An ABC of group analysis, Novoe v Zhizni, Nauke, Tekhnike. Seriya Matematika, Kibernetika, no. 8, Znanie, Moscow, 1989 (in Russian).
    Ibragimov N.H., Introduction to modern group analysis, Tau, Ufa, 2000 (revised edition in English).
  4. Ibragimov N.H., Elementary Lie group analysis and ordinary differential equations, J. Wiley, Chichester, 1999.
  5. Winternitz P., Lie groups and solutions of nonlinear differential equations, in Nonlinear Phenomena, Editor K.B. Wolf, Lecture Notes in Physics, Vol. 189, Springer-Verlag, New York, 1983, 263-305.
  6. Cariñena J.F., Grabowski J., Marmo G., Lie-Scheffers systems: a geometric approach, Bibliopolis, Napoli, 2000.
  7. Cariñena J.F., Grabowski J., Ramos A., Reduction of time-dependent systems admitting a superposition principle, Acta Appl. Math. 66 (2001), 67-87.
  8. Cariñena J.F., Grabowski J., Marmo G., Some applications in physics of differential equation systems admitting a superposition rule, Rep. Math. Phys. 48 (2001), 47-58.
  9. Anderson R.L., A nonlinear superposition principle admitted by coupled Riccati equations of the projective type, Lett. Math. Phys. 4 (1980), 1-7.
  10. Harnad J., Winternitz P., Anderson R.L., Superposition principles for matrix Riccati equations, J. Math. Phys. 24 (1983), 1062-1072.
  11. del Olmo M.A., Rodríguez M.A., Winternitz P., Simple subgroups of simple Lie groups and nonlinear differential equations with superposition principles, J. Math. Phys. 27 (1986), 14-23.
  12. Kevrekidis P.G., Drossinos Y., Nonlinearity from linearity: the Ermakov-Pinney equation revisited, Math. Comput. Simulation 74 (2007), 196-202.
  13. Leach P.G.L., Karasu A., Nucci M.C., Andriopoulos K., Ermakov's superintegrable toy and nonlocal symmetries, SIGMA 1 (2005), 018, 15 pages, nlin.SI/0511055.
  14. del Olmo M.A., Rodríguez M.A., Winternitz P., Superposition formulas for rectangular matrix Riccati equations, J. Math. Phys. 28 (1987), 530-535.
  15. Cariñena J.F., Ramos A., Riccati equation, factorization method and shape invariance, Rev. Math. Phys. 12 (2000), 1279-1304, math-ph/9910020.
  16. Cariñena J.F., Ramos A., A new geometric approach to Lie systems and physical applications, Acta Appl. Math. 70 (2002), 43-69, math-ph/0110023.
  17. Cariñena J.F., Marmo G., Nasarre J., The nonlinear superposition principle and the Wei-Norman method, Internat. J. Modern Phys. A 13 (1998), 3601-3627, physics/9802041.
  18. Cariñena J.F., Grabowski J., Marmo G., Superposition rules, Lie theorem and partial differential equations, Rep. Math. Phys. 60 (2007), 237-258 math-ph/0610013.
  19. Cariñena J.F., de Lucas J., Rañada M.F., Nonlinear superposition rules and Ermakov systems, in Differential Geometric Methods in Mechanics and Field Theory, Editors F. Cantrijn, M. Crampin and B. Langerock, Academia Press, 2007, 15-33.
  20. Milne W.E., The numerical determination of characteristic numbers, Phys. Rev. 35 (1930), 863-67.
  21. Pinney E., The nonlinear differential equation y"+p(x)y+cy-3=0, Proc. Amer. Math. Soc. 1 (1950), 681.
  22. Hawkins R.M., Lidsey J.E., Ermakov-Pinney equation in scalar field cosmologies, Phys. Rev. D 66 (2002), 023523, 8 pages, astro-ph/0112139.
  23. Lidsey L.E., Cosmic dynamics of Bose-Einstein condensates, Classical Quantum Gravity 21 (2004), 777-785, gr-qc/0307037.
  24. Haas F., Anisotropic Bose-Einstein condensates and completely integrable dynamical systems, Phys. Rev. A 65 (2002), 033603, 6 pages, cond-mat/0211353.
  25. Fernández Guasti M., Moya-Cessa H., Amplitude and phase representation of quantum invariants for the time-dependent harmonic osicllator, Phys. Rev. A 67 (2003), 063803, 5 pages, quant-ph/0212073.
  26. Gauthier S., An exact invariant for the time dependent double well anharmonic oscillators: Lie theory and quasi-invariance groups, J. Phys. A: Math. Gen. 17 (1984), 2633-2639.
  27. Ray J.R., Reid J.L., More exact invariants for the time-dependent harmonic oscillator, Phys. Lett. A 71 (1979), 317-318.
  28. Dhara A.K., Lawande S.V., Time-dependent invariants and the Feynman propagator, Phys. Rev. A 30 (1984), 560-567.
  29. Lewis H.R., Classical and quantum systems with time-dependent harmonic-oscillator-type Hamiltonians, Phys. Rev. Lett. 18 (1967), 510-512.
  30. Reid J.L., Ray J.R., Ermakov systems, Noether's theorem and the Sarlet-Bahar method, Lett. Math. Phys. 4 (1980), 235-240.
  31. Cerveró J.M., Lejarreta J.D., Ermakov Hamiltonians, Phys. Lett. A 156 (1991), 201-205.
  32. Sarlet W., Exact invariants for time-dependent Hamiltonian systems with one degree of freedom, J. Phys. A: Math. Gen. 11 (1978), 843-854.
  33. Govinder K.S., Athorne C., Leach P.G.L., The algebraic structure of generalized Ermakov systems in three dimensions, J. Phys. A: Math. Gen. 26 (1993), 4035-4046.
  34. Athorne C., Rogers C., Ramgulam U., Osbaldestin A., On linearization of the Ermakov system, Phys. Lett. A 143 (1990), 207-212.
  35. Sarlet W., Cantrijn F., A generalization of the nonlinear superposition idea for Ermakov systems, Phys. Lett. A 88 (1982), 383-387.
  36. Reid J.L., Ray J.R., Ermakov systems, nonlinear superposition and solutions of nonlinear equations of motion, J. Math. Phys. 21 (1980), 1583-1587.
  37. Govinder K.S., Leach P.G.L., Ermakov systems: a group theoretic approach, Phys. Lett. A 186 (1994), 391-395.
  38. Athorne C., Projective lifts and generalised Ermakov and Bernoulli systems, J. Math. Anal. Appl. 233 (1999), 552-563.
  39. Rogers C., Schief W.K., Bassom A., Ermakov systems with arbitrary order, dimension. Structure and linearisation, J. Phys. A: Math. Gen. 29 (1996), 903-911.
  40. Leach P.G.L., Generalized Ermakov systems, Phys. Lett. A 158 (1991), 102-106.
  41. Sarlet W., Further generalization of Ray-Reid systems, Phys. Lett. A 82 (1981), 161-164.
  42. Calogero F., Solution of a three body problem in one dimension, J. Math. Phys. 10 (1969), 2191-2196.
  43. Perelomov A.M., Integrable systems of classical mechanics and Lie algebras, Birkhäuser Verlag, Basel, 1990.
  44. Chalykh O.A., Vesselov A.P., A remark on rational isochronous potentials, J. Nonlinear Math. Phys. 12 (2005), suppl. 1, 179-183, math-ph/0409062.
  45. Asorey M., Cariñena J.F., Marmo G., Perelomov A.M., Isoperiodic classical systems and their quantum counterparts, Ann. Phys. 322 (2007), 1444-1465, arXiv:0707.4465.
  46. Cariñena J.F., Perelomov A.M., Rañada M.F., Isochronous classical systems and quantum systems with equally spaced spectra, in Particles and Fields: Classical and Quantum, J. Phys. Conf. Ser. 87 (2007), 012007, 14 pages.
  47. Leach P.G.L., Karasu A., The Lie algebra sl(2,R) and so-called Kepler-Ermakov systems, J. Nonlinear Math. Phys. 11 (2004), 269-275.
  48. Karasu A., Yildrim H., On the Lie symmetries of the Kepler-Ermakov systems, J. Nonlinear Math. Phys. 9 (2002), 475-482, math-ph/0306037.
  49. Ray J.R., Reid J.L., Exact time-dependent invariants for N-dimensional systems, Phys. Lett. A 74 (1979), 23-25.
  50. Ray J.R., Invariants for nonlinear equations of motion, Progr. Theoret. Phys. 65 (1981), 877-882.
  51. Cariñena J.F., Ramos A., Integrability of the Riccati equation from a group theoretical viewpoint, Internat. J. Modern Phys. A 14 (1999), 1935-1951, math-ph/9810005.
  52. Cariñena J.F., de Lucas J., Ramos A., A geometric approach to integrability conditions for Riccati equations, Electron. J. Differential Equations 2007 (2007), 122, 14 pages.

Previous article   Next article   Contents of Volume 4 (2008)