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Canonoid transformations and master symmetries
1. | Departamento de Física Teórica and IUMA, Facultad de Ciencias, Universidad de Zaragoza, Pedro Cerbuna 12, 50.009, Zaragoza |
2. | Departamento de Física Teórica, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain, Spain |
References:
[1] |
R. Abraham and J. E. Marsden, "Foundations of Mechanics," Second edition, revised and enlarged, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. |
[2] |
G. Marmo, E. J. Saletan, A. Simoni and B. Vitale, "Dynamical Systems: A Differential Geometric Approach to Symmetry and Reduction," A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1985. |
[3] |
M. Crampin and F. A. E. Pirani, "Applicable Differential Geometry,'' London Mathematical Society Lecture Note Series, 59, Cambridge Univ. Press, Cambridge, 1986. |
[4] |
M. de León and P. R. Rodrigues, "Methods of Differential Geometry in Analytical Mechanics,'' North-Holland Mathematics Studies, 158, North-Holland Publishing Co., Amsterdam, 1989. |
[5] |
G. Morandi, C. Ferrario, G. Lo Vecchio, G. Marmo and C. Rubano, The inverse problem in the calculus of variations and the geometry of the tangent bundle, Phys. Rep., 188 (1990), 147-284.
doi: 10.1016/0370-1573(90)90137-Q. |
[6] |
E. J. Saletan and A. H. Cromer, "Theoretical Mechanics,'' John Wiley & Sons, 1971. |
[7] |
E. J. Saletan and J. V. José, "Classical Mechanics: A Contemporary Approach,'' Cambridge Univ. Press, Cambridge, 1998. |
[8] |
J. F. Cariñena and M. F. Rañada, Canonoid transformations from a geometric perspective, J. Math. Phys., 29 (1988), 2181-2186.
doi: 10.1063/1.528146. |
[9] |
F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys., 19 (1978), 1156-1162.
doi: 10.1063/1.523777. |
[10] |
J. F. Cariñena and L. A. Ibort, Non-Noether constants of motion, J. Phys., 16 (1983), 1-7.
doi: 10.1088/0305-4470/16/1/010. |
[11] |
C. López, E. Martínez and M. F. Rañada, Dynamical symmetries, non-Cartan symmetries and superintegrability of the $n$-dimensional harmonic oscillator, J. Phys. A, 32 (1999), 1241-1249.
doi: 10.1088/0305-4470/32/7/013. |
[12] |
J. F. Cariñena, F. Falceto and M. F. Rañada, A geometric approach to a generalised Virial theorem, J. Phys. A, 45 (2012), 395210, 19 pp.
doi: 10.1088/1751-8113/45/39/395210. |
[13] |
G. Landolfi and G. Soliani, On certain canonoid transformations and invariants for the parametric oscillator, J. Phys. A, 40 (2007), 3413-3423
doi: 10.1088/1751-8113/40/13/009. |
[14] |
P. Tempesta, E. Alfinito, R. A. Leo and G. Soliani, Quantum models related to fouled Hamiltonians of the harmonic oscillator, J. Math. Phys., 43 (2002), 3538-3553.
doi: 10.1063/1.1479300. |
[15] |
T. Dereli, A. Teǧmen and T. Hakioǧlu, Canonical transformations in three-dimensional phase-space, Int. J. Modern Phys. A, 24 (2009), 4769-4788.
doi: 10.1142/S0217751X09044760. |
[16] |
B. Nachtergaele and A. Verbeure, Groups of canonical transformatioins and the virial-Noether theorem, J. Geom. Phys., 3 (1986), 315-325.
doi: 10.1016/0393-0440(86)90012-4. |
[17] |
C. Leubner and M. Marte, Generalized canonical transformations and constants of the motion, Phys. Lett. A, 101 (1984), 179-181.
doi: 10.1016/0375-9601(84)90372-4. |
[18] |
L. Negri, L. C. Oliveira and J. M. Teixeira, Canonoid transformations and constants of motion, J. Math. Phys., 28 (1987), 2369-2372.
doi: 10.1063/1.527772. |
[19] |
D. G. Currie and E. J. Saletan, Canonical transformations and quadratic Hamiltonians, Nuovo Cimento B (11), 9 (1972), 143-153.
doi: 10.1007/BF02735514. |
[20] |
J. F. Cariñena and M. F. Rañada, Generating functions, bi-Hamiltonian systems, and the quadratic-Hamiltonian theorem, J. Math. Phys., 31 (1990), 801-807.
doi: 10.1063/1.529028. |
[21] |
J. F. Cariñena, J. M. Gracia-Bondía, L. A. Ibort, C. López and J. C. Várilly, Distinguished Hamiltonian theorem for homogeneous symplectic manifolds, Lett. Math. Phys., 23 (1991), 35-44.
doi: 10.1007/BF01811292. |
[22] |
R. Schmid, The quadratic-Hamiltonian theorem in infinite dimensions, J. Math. Phys., 29 (1988), 2010-2011.
doi: 10.1063/1.527858. |
[23] |
P. A. Damianou, Symmetries of Toda equations, J. Phys. A, 26 (1993), 3791-3796.
doi: 10.1088/0305-4470/26/15/027. |
[24] |
R. L. Fernandes, On the master symmetries and bi-Hamiltonian structure of the Toda lattice, J. Phys. A, 26 (1993), 3797-3803.
doi: 10.1088/0305-4470/26/15/028. |
[25] |
M. F. Rañada, Superintegrability of the Calogero-Moser system: Constants of motion, master symmetries, and time-dependent symmetries, J. Math. Phys., 40 (1999), 236-247.
doi: 10.1063/1.532770. |
[26] |
R. G. Smirnov, On the master symmetries related to certain classes of integrable Hamiltonian systems, J. Phys. A, 29 (1996), 8133-8138.
doi: 10.1088/0305-4470/29/24/034. |
[27] |
F. Finkel and A. S. Fokas, On the construction of evolution equations admitting a master symmetry, Phys. Lett. A, 293 (2002), 36-44.
doi: 10.1016/S0375-9601(01)00836-2. |
[28] |
R. Caseiro, Master integrals, superintegrability and quadratic algebras, Bull. Sci. Math., 126 (2002), 617-630.
doi: 10.1016/S0007-4497(02)01117-X. |
[29] |
P. A. Damianou and Ch. Sophocleous, Noether and master symmetries for the Toda lattice, Appl. Math. Lett., 18 (2005), 163-170.
doi: 10.1016/j.aml.2004.02.005. |
[30] |
M. F. Rañada, Master symmetries, non-Hamiltonian symmetries and superintegrability of the generalized Smoridinsky-Winternitz system, J. Phys. A, 45 (2012), 145204, 13 pp.
doi: 10.1088/1751-8113/45/14/145204. |
[31] |
J. F. Cariñena and L. A. Ibort, Noncanonical groups of transformations, anomalies, and cohomology, J. Math. Phys., 29 (1988), 541-545.
doi: 10.1063/1.528047. |
[32] |
R. Abraham, J. E. Marsden and T. Ratiu, "Manifolds, Tensor Analysis, and Applications,'' Second edition, Applied Mathematical Sciences, 75, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1029-0. |
show all references
References:
[1] |
R. Abraham and J. E. Marsden, "Foundations of Mechanics," Second edition, revised and enlarged, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. |
[2] |
G. Marmo, E. J. Saletan, A. Simoni and B. Vitale, "Dynamical Systems: A Differential Geometric Approach to Symmetry and Reduction," A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1985. |
[3] |
M. Crampin and F. A. E. Pirani, "Applicable Differential Geometry,'' London Mathematical Society Lecture Note Series, 59, Cambridge Univ. Press, Cambridge, 1986. |
[4] |
M. de León and P. R. Rodrigues, "Methods of Differential Geometry in Analytical Mechanics,'' North-Holland Mathematics Studies, 158, North-Holland Publishing Co., Amsterdam, 1989. |
[5] |
G. Morandi, C. Ferrario, G. Lo Vecchio, G. Marmo and C. Rubano, The inverse problem in the calculus of variations and the geometry of the tangent bundle, Phys. Rep., 188 (1990), 147-284.
doi: 10.1016/0370-1573(90)90137-Q. |
[6] |
E. J. Saletan and A. H. Cromer, "Theoretical Mechanics,'' John Wiley & Sons, 1971. |
[7] |
E. J. Saletan and J. V. José, "Classical Mechanics: A Contemporary Approach,'' Cambridge Univ. Press, Cambridge, 1998. |
[8] |
J. F. Cariñena and M. F. Rañada, Canonoid transformations from a geometric perspective, J. Math. Phys., 29 (1988), 2181-2186.
doi: 10.1063/1.528146. |
[9] |
F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys., 19 (1978), 1156-1162.
doi: 10.1063/1.523777. |
[10] |
J. F. Cariñena and L. A. Ibort, Non-Noether constants of motion, J. Phys., 16 (1983), 1-7.
doi: 10.1088/0305-4470/16/1/010. |
[11] |
C. López, E. Martínez and M. F. Rañada, Dynamical symmetries, non-Cartan symmetries and superintegrability of the $n$-dimensional harmonic oscillator, J. Phys. A, 32 (1999), 1241-1249.
doi: 10.1088/0305-4470/32/7/013. |
[12] |
J. F. Cariñena, F. Falceto and M. F. Rañada, A geometric approach to a generalised Virial theorem, J. Phys. A, 45 (2012), 395210, 19 pp.
doi: 10.1088/1751-8113/45/39/395210. |
[13] |
G. Landolfi and G. Soliani, On certain canonoid transformations and invariants for the parametric oscillator, J. Phys. A, 40 (2007), 3413-3423
doi: 10.1088/1751-8113/40/13/009. |
[14] |
P. Tempesta, E. Alfinito, R. A. Leo and G. Soliani, Quantum models related to fouled Hamiltonians of the harmonic oscillator, J. Math. Phys., 43 (2002), 3538-3553.
doi: 10.1063/1.1479300. |
[15] |
T. Dereli, A. Teǧmen and T. Hakioǧlu, Canonical transformations in three-dimensional phase-space, Int. J. Modern Phys. A, 24 (2009), 4769-4788.
doi: 10.1142/S0217751X09044760. |
[16] |
B. Nachtergaele and A. Verbeure, Groups of canonical transformatioins and the virial-Noether theorem, J. Geom. Phys., 3 (1986), 315-325.
doi: 10.1016/0393-0440(86)90012-4. |
[17] |
C. Leubner and M. Marte, Generalized canonical transformations and constants of the motion, Phys. Lett. A, 101 (1984), 179-181.
doi: 10.1016/0375-9601(84)90372-4. |
[18] |
L. Negri, L. C. Oliveira and J. M. Teixeira, Canonoid transformations and constants of motion, J. Math. Phys., 28 (1987), 2369-2372.
doi: 10.1063/1.527772. |
[19] |
D. G. Currie and E. J. Saletan, Canonical transformations and quadratic Hamiltonians, Nuovo Cimento B (11), 9 (1972), 143-153.
doi: 10.1007/BF02735514. |
[20] |
J. F. Cariñena and M. F. Rañada, Generating functions, bi-Hamiltonian systems, and the quadratic-Hamiltonian theorem, J. Math. Phys., 31 (1990), 801-807.
doi: 10.1063/1.529028. |
[21] |
J. F. Cariñena, J. M. Gracia-Bondía, L. A. Ibort, C. López and J. C. Várilly, Distinguished Hamiltonian theorem for homogeneous symplectic manifolds, Lett. Math. Phys., 23 (1991), 35-44.
doi: 10.1007/BF01811292. |
[22] |
R. Schmid, The quadratic-Hamiltonian theorem in infinite dimensions, J. Math. Phys., 29 (1988), 2010-2011.
doi: 10.1063/1.527858. |
[23] |
P. A. Damianou, Symmetries of Toda equations, J. Phys. A, 26 (1993), 3791-3796.
doi: 10.1088/0305-4470/26/15/027. |
[24] |
R. L. Fernandes, On the master symmetries and bi-Hamiltonian structure of the Toda lattice, J. Phys. A, 26 (1993), 3797-3803.
doi: 10.1088/0305-4470/26/15/028. |
[25] |
M. F. Rañada, Superintegrability of the Calogero-Moser system: Constants of motion, master symmetries, and time-dependent symmetries, J. Math. Phys., 40 (1999), 236-247.
doi: 10.1063/1.532770. |
[26] |
R. G. Smirnov, On the master symmetries related to certain classes of integrable Hamiltonian systems, J. Phys. A, 29 (1996), 8133-8138.
doi: 10.1088/0305-4470/29/24/034. |
[27] |
F. Finkel and A. S. Fokas, On the construction of evolution equations admitting a master symmetry, Phys. Lett. A, 293 (2002), 36-44.
doi: 10.1016/S0375-9601(01)00836-2. |
[28] |
R. Caseiro, Master integrals, superintegrability and quadratic algebras, Bull. Sci. Math., 126 (2002), 617-630.
doi: 10.1016/S0007-4497(02)01117-X. |
[29] |
P. A. Damianou and Ch. Sophocleous, Noether and master symmetries for the Toda lattice, Appl. Math. Lett., 18 (2005), 163-170.
doi: 10.1016/j.aml.2004.02.005. |
[30] |
M. F. Rañada, Master symmetries, non-Hamiltonian symmetries and superintegrability of the generalized Smoridinsky-Winternitz system, J. Phys. A, 45 (2012), 145204, 13 pp.
doi: 10.1088/1751-8113/45/14/145204. |
[31] |
J. F. Cariñena and L. A. Ibort, Noncanonical groups of transformations, anomalies, and cohomology, J. Math. Phys., 29 (1988), 541-545.
doi: 10.1063/1.528047. |
[32] |
R. Abraham, J. E. Marsden and T. Ratiu, "Manifolds, Tensor Analysis, and Applications,'' Second edition, Applied Mathematical Sciences, 75, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1029-0. |
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