A non-linear oscillator with quasi-harmonic behaviour: Two- and n-dimensional oscillators

Nonlinearity

Volumn 17, Issue 5, 2004, Pages 1941-1963

  Cariñena, José F ^a   Rañada, Manuel F ^a   Santander, Mariano ^b   Senthilvelan, Murugaian ^c  

^a UNIVERSITY OF ZARAGOZA   (Spain)

^b UNIVERSITY OF VALLADOLID   (Spain)

^c BHARATHIDASAN UNIVERSITY   (India)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 4644336646     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/17/5/019     Document Type: Article

References (44)

1. Mathews P M and Lakshmanan M 1974 On a unique non-linear oscillator Q. Appl. Math. 32 215-8
2. Lakshmanan M and Rajasekar S 2003 Nonlinear dynamics Integrability, Chaos and Patterns, Advanced Texts in Physics (Berlin: Springer)
3. Delbourgo R, Salam A and Strathdee J 1969 Infinities of non-linear and Lagrangian theories Phys. Rev. 187 1999-2007
4. Nishijima K and Watanabe T 1972 Green's functions in non-linear field theories Prog. Theor. Phys. 47 996-1003
5. 2m anharmonic oscillators J. Math. Phys. 14 1190-5
6. Mathews P M and Lakshmanan M 1975 A quantum mechanically solvable nonpolynomial Lagrangian with velocity-dependent interaction Nuovo Cimento A 26 299-315
7. 2) J. Math. Phys. 19 2018-22
8. 2) J. Phys. A: Math. Gen. 19 L253-8
9. 2) potential J. Math. Phys. 21 2780-5
10. 2) interaction Phys. Lett. A 83 121-2
11. Lakshmanan M and Eswaran K 1975 Quantum dynamics of a solvable nonlinear chiral model J. Phys. A: Math. Gen. 8 1658-69
12. 2) in two and three dimensions Hadron. J. 13 47-56
13. Lin H E, Lin W C and Sugano R 1970 On velocity dependent potentials in quantum mechanics Nucl. Phys. B 16 431-49
14. Velo G and Wess J 1971 A solvable quantum mechanical model with non-linear transformations laws Nuovo Cimenta A 1 177-87
15. Lévy-Leblond J M 1995 Position-dependent effective mass and Galilean invariance Phys. Rev. A 52 1485-9
16. Rañada M F and Santander M 2002 On some properties of the harmonic oscillator on spaces of constant curvature Rep. Math. Phys. 49 335-43
17. 2 J. Math. Phys. 43 431-51
18. 2 II J. Math. Phys. 44 2149-67
19. Sarlet W 1991 New aspects of integrability of generalized Hénon-Heiles systems J. Phys. A: Math. Gen. 24 5245-51
20. Ravoson V, Gavrilov L and Caboz R 1993 Separability and Lax pairs for the Hénon-Heiles system J. Math. Phys. 34 2385-93
21. Rauch-Wojciechowski S and Tsiganov A V 1996 Quasi-point separation of variables for the Hénon-Heiles system and a system with a quartic potential J. Phys. A.: Math. Gen. 29 7769-78
22. Cariñena J F and Rañada M F 1999 Helmholtz conditions and alternative Lagrangians: study of an integrable Hénon-Heiles system Int. J. Theor. Phys. 38 2049-61
23. Perelomov A M 1990 Integrable Systems of Classical Mechanics and Lie algebras (Basle: Birkhauser)
24. Cariñena J F, Marmo G and Rañada M F 2002 Non-symplectic symmetries and bi-Hamiltonian structures of the rational harmonic oscillator J. Phys. A: Math. Gen. 35 L679-86
25. Fris T I, Mandrosov V, Smorodinsky Y A, Uhlir M and Winternitz P 1965 On higher symmetries in quantum mechanics Phys. Lett. 16 354-6
26. Evans N W 1990 Superintegrability in classical mechanics Phys. Rev. A 41 5666-76
27. Grosche C, Pogosyan G S and Sissakian A N 1995 Path integral discussion for Smorodinsky-Winternitz potentials Fortschr. Phys. 43 453-521
28. Rañada M F 1997 Superintegrable n = 2 systems, quadratic constants of motion, and potentials of Drach J. Math. Phys. 38 4165-78
29. Benenti S, Chanu C and Rastelli G 2000 The super-separability of the three-body inverse-square Calogero system J. Math. Phys. 41 4654-78
30. Tempesta P, Turbiner A V and Winternitz P 2001 Exact solvability of super-integrable systems J. Math. Phys. 42 4248-57
31. Ballesteros A, Herranz F J, Santander M and Sanz-Gil T 2003 Maximal super-integrability on N-dimensional curved spaces J. Phys. A: Math. Gen. 36 L93-9
32. Higgs P W 1979 Dynamical symmetries in a spherical geometry I J. Phys. A: Math. Gen. 12 309-23
33. Leemon H I 1979 Dynamical symmetries in a spherical geometry II J. Phys. A: Math. Gen. 12 489-501
34. Grosche C, Pogosyan G S and Sissakian A N 1995 Path integral discussion for Smorodinsky-Winternitz potentials II Fortschr, Phys. 43 523-63
35. 2 J. Math. Phys. 40 5026-57
36. Slawianowski J J 2000 Bertrand systems on spaces of constant sectional curvature Rep. Math. Phys. 46 429-60
37. Kalnins E G, Kress J M, Pogosyan G S and Miller W 2001 Completeness of super-integrability in two-dimensional constant-curvature spaces J. Phys. A: Math. Gen. 34 4705-20
38. Kalnins E G, Kress J M and Winternitz P 2002 Super-integrability in a two-dimensional space of nonconstant curvature J. Math. Phys. 43 970-83
39. Kalnins E G, Miller W and Pogosyan G S 2002 The Coulomb-oscillator relation on n-dimensional spheres and hyperboloids Phys. At. Nuc. 65 1119-27
40. 2 Classical and Quantum Integrability (Banach Center Publ.) vol 59 (Warsaw: Polish Acad. Sci.) pp 243-55
41. Scarf F L 1958 New soluble energy band problem Phys. Rev. 112 1137-40
42. De R, Dutt R and Sukhatme U 1992 Path-integral solutions for shape-invariance potentials using point canonical transformations Phys. Rev. A 46 6869-80
43. Grosche C 1993 Path integral discussion for Scarf-like potentials Nuovo Cimenta B 108 1365-76
44. Daskaloyannis C 2001 Quadratic Poisson algebras of two-dimensional classical super-integrable systems and quadratic associative algebras of quantum super-integrable systems J. Math. Phys. 42 1100-19