American Institute of Mathematical Sciences

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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.

[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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