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June  2013, 5(2): 151-166. doi: 10.3934/jgm.2013.5.151

Canonoid transformations and master symmetries

José F. Cariñena 1, , Fernando Falceto 2, and  Manuel F. Rañada 2,

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

Received  December 2012 Revised  March 2013 Published  July 2013

Different types of transformations of a dynamical system, that are compatible with the Hamiltonian structure, are discussed making use of a geometric formalism. Firstly, the case of canonoid transformations is studied with great detail and then the properties of master symmetries are also analyzed. The relations between the existence of constants of motion and the properties of canonoid symmetries is discussed making use of a family of boundary and coboundary operators.
Keywords: master symmetry, Canonoid transformation, constants of motion..
Mathematics Subject Classification: Primary: 70H15, 53D22; Secondary: 37J05, 37J1.
Citation: José F. Cariñena, Fernando Falceto, Manuel F. Rañada. Canonoid transformations and master symmetries. Journal of Geometric Mechanics, 2013, 5 (2) : 151-166. doi: 10.3934/jgm.2013.5.151
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.  Google Scholar

[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.  Google Scholar

[3]

M. Crampin and F. A. E. Pirani, "Applicable Differential Geometry,'' London Mathematical Society Lecture Note Series, 59, Cambridge Univ. Press, Cambridge, 1986.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

E. J. Saletan and A. H. Cromer, "Theoretical Mechanics,'' John Wiley & Sons, 1971. Google Scholar

[7]

E. J. Saletan and J. V. José, "Classical Mechanics: A Contemporary Approach,'' Cambridge Univ. Press, Cambridge, 1998.  Google Scholar

[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.  Google Scholar

[9]

F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys., 19 (1978), 1156-1162. doi: 10.1063/1.523777.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

R. Schmid, The quadratic-Hamiltonian theorem in infinite dimensions, J. Math. Phys., 29 (1988), 2010-2011. doi: 10.1063/1.527858.  Google Scholar

[23]

P. A. Damianou, Symmetries of Toda equations, J. Phys. A, 26 (1993), 3791-3796. doi: 10.1088/0305-4470/26/15/027.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

R. Caseiro, Master integrals, superintegrability and quadratic algebras, Bull. Sci. Math., 126 (2002), 617-630. doi: 10.1016/S0007-4497(02)01117-X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

M. Crampin and F. A. E. Pirani, "Applicable Differential Geometry,'' London Mathematical Society Lecture Note Series, 59, Cambridge Univ. Press, Cambridge, 1986.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

E. J. Saletan and A. H. Cromer, "Theoretical Mechanics,'' John Wiley & Sons, 1971. Google Scholar

[7]

E. J. Saletan and J. V. José, "Classical Mechanics: A Contemporary Approach,'' Cambridge Univ. Press, Cambridge, 1998.  Google Scholar

[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.  Google Scholar

[9]

F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys., 19 (1978), 1156-1162. doi: 10.1063/1.523777.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

R. Schmid, The quadratic-Hamiltonian theorem in infinite dimensions, J. Math. Phys., 29 (1988), 2010-2011. doi: 10.1063/1.527858.  Google Scholar

[23]

P. A. Damianou, Symmetries of Toda equations, J. Phys. A, 26 (1993), 3791-3796. doi: 10.1088/0305-4470/26/15/027.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

R. Caseiro, Master integrals, superintegrability and quadratic algebras, Bull. Sci. Math., 126 (2002), 617-630. doi: 10.1016/S0007-4497(02)01117-X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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