Applications of lie systems in dissipative milne-pinney equations

International Journal of Geometric Methods in Modern Physics

Volumn 6, Issue 4, 2009, Pages 683-699

  Cariñena, José F ^a   de Lucas, Javier ^a  

^a UNIVERSITY OF ZARAGOZA   (Spain)

Author keywords

Ermakov system; Milne Pinney equation; Quasi Lie scheme; Superposition rule

Indexed keywords

EID: 67650556416     PISSN: 02198878     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0219887809003758     Document Type: Article

References (28)

1. W. E. Milne, The numerical determination of characteristic numbers, Phys. Rev. 35 (1930) 863-867.
2. 3 = 0, Proc. A.M.S. 1 (1950) 681.
3. V. P. Ermakov, Second-order differential equations. Conditions of complete integrability, Univ. Isz. Kiev Series III 9, 1-25 (1880) (translation by A. O. Harin). See: Appl. Anal. Discrete Math. 2 (2008) 123-148.
4. J. F. Cariñena and J. de Lucas, A nonlinear superposition rule for solutions of Milne-Pinney equation, Phys. Lett. A 372 (2008) 5385-5389.
5. F. Haas, Approximate solution for a damped Pinney equation, math-ph/ 0712.4083.
6. Z. E. Musielak, Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients, J. Phys. A: Math. Theor. 41 (2008) 055205.
7. Z. E. Musielak, D. Roy and L. D. Swift, Method to derive Lagrangian and Hamiltonian for a nonlinear dynamical system with variable coefficients, Chaos, Solitons and Fractals 38 (2008) 894-902.
8. S.-W. Qian, B.-W. Huang and Z.-Y. Gu, Ermakov invariant and the general solution for a damped harmonic oscillator with a force quadratic in velocity, J. Phys. A 34 (2001) 5613-5617.
9. S. Lie and G. Scheffers, Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen, Edited and revised by G. Scheffers (Teubner, Leipzig, 1893).
10. J. F. Cariñena, J. Grabowski and G. Marmo, Lie-Scheffers Systems: A Geometric Approach (Bibliopolis, Napoli, 2000).
11. J. F. Cariñena and A. Ramos, A new geometric approach to Lie systems and physical applications, Acta Appl. Math. 70 (2002) 43-69.
12. J. F. Cariñena, J. Grabowski and G. Marmo, Superposition rules, Lie theorem and partial differential equations, Rep. Math. Phys. 60 (2007) 237-258.
13. J. F. Cariñena, J. de Lucas and M. F. Rañada, Recent applications of the theory of Lie systems in Ermakov systems, SIGMA 4 (2008) 031.
14. J. F. Cariñena, J. Grabowski and J. de Lucas, Quasi-Lie schemes: Theory and applications, arXiv: 0810.1160.
15. I. A. Pedrosa, Canonical transformations and exact invariants for dissipative systems, J. Math. Phys. 28 (1987) 2662-2664.
16. R. Milson, Liouville transformation and exactly solvable Schrödinger equation, Int. J. Theor. Phys. 37 (1998) 1735-1752.
17. L. M. Berkovich, Transformation of Sturm-Liouville differential equations, Funct. Anal. Appl. 16 (1982) 190-192.
18. T. I. Brazier and P. G. L. Leach, Invariants for dissipative systems and Noether's theorem, Rev. Mex. Fis. 40 (1994) 378-385.
19. J. F. Cariñena, J. de Lucas and A. Ramos, A geometric approach to time evolution operators of Lie quantum systems, to appear in Int. J. Theor. Phys. 48 (2009).
20. J. M. Cerveró and J. Villarroel, On the quantum theory of the damped harmonic oscillator, J. Phys. A: Math. Gen. 17 (1984) 2963-2971.
21. J. F. Cariñena, J. de Lucas and M. F. Rañada, Nonlinear superpositions and Ermakov systems, in: Differential Geometric Methods in Mechanics and Field Theory, eds. F. Cantrijn, M. Crampin and B. Langerock (Academia Press, Gent, 2007), pp. 15-33.
22. A. Nassar, Ermakov and non-Ermakov systems in quantum dissipative models, J. Math. Phys. 27 (1986) 755-758.
23. M. Aktas and R. Sever, Effective mass Schrödinger equation for exactly solvable class of one-dimensional potentials, J. Math. Chem. 43 (2008) 92-100.
24. O. von Roos, Position-dependent effective mass in semiconductor theory, Phys. Rev. B 27 (1983) 7547-7552.
25. R. Redheffer, Steen's equation and its generalizations, Aequationes Math. 58 (1999) 60-72.
26. M. Chini, Sull'equazione del 2° ordine lineare omogenea, Atti. Accad. Sci. Torino 33 (1898) 737-745.
27. J. Walter, Bemerkungen zu dem Grenzpunktfallkriterium von N. Levinson, Math. Z. 105 (1968) 345-350.
28. R. K. Colegrave and M. S. Abdalla, Invariants for the time-dependent harmonic oscillators, J. Phys. A: Math. Gen. 12 (1983) 3805-3815.