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September  2020, 12(3): 541-551. doi: 10.3934/jgm.2020009

A summary on symmetries and conserved quantities of autonomous Hamiltonian systems

Narciso Román-Roy

Departament of Mathematics. Universidad Politécnica de Cataluña, Edificio C-3, Campus Norte UPC. C/ Jordi Girona 1. 08034 Barcelona, Spain

Received  April 2019 Revised  December 2019 Published  March 2020

A complete geometric classification of symmetries of autonomous Hamiltonian systems is established; explaining how to obtain their associated conserved quantities in all cases. In particular, first we review well-known results and properties about the symmetries of the Hamiltonian and of the symplectic form and then some new kinds of non-symplectic symmetries and their conserved quantities are introduced and studied.

Keywords: Symmetries, conserved quantities, Hamiltonian systems, Noether theorem, symplectic manifolds.
Mathematics Subject Classification: Primary: 37K05, 70H05, 70S10; Secondary: 37J15, 53D05, 70S05.
Citation: Narciso Román-Roy. A summary on symmetries and conserved quantities of autonomous Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (3) : 541-551. doi: 10.3934/jgm.2020009
References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, Second edition, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. doi: 10.1090/chel/364.  Google Scholar

[2]

A. Arancibia and M. S. Plyushchay, Chiral asymmetry in propagation of soliton defects in crystalline backgrounds, Phys. Rev. D, 92 (2015), 105009, 20 pp. doi: 10.1103/PhysRevD.92.105009.  Google Scholar

[3]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[4]

P. Birtea and R. M. Tudoran, Non-Noether conservation laws, Int. J. Geom. Methods Mod. Phys., 9 (2012), 1220004, 5 pp. doi: 10.1142/S0219887812200046.  Google Scholar

[5]

A. V. Bolsinov, Compatible Poisson brackets on Lie algebras and the completeness of families of functions in involution, Math. USSR-Izvestiya, 38 (1992), 69-90.  doi: 10.1070/IM1992v038n01ABEH002187.  Google Scholar

[6]

A. V. Bolsinov and A. V. Borisov, Compatible Poisson brackets on Lie algebras, Math. Notes, 72 (2002), 10-30.  doi: 10.1023/A:1019856702638.  Google Scholar

[7]

J. F. Cariñena and L. A. Ibort, Non-Noether constants of motion, J. Phys. A, 16 (1983), 1-7.  doi: 10.1088/0305-4470/16/1/010.  Google Scholar

[8]

J. F. Cariñena, G. Marmo and M. F. Rañada, Non-symplectic symmetries and bi-Hamiltonian structures of the rational harmonic oscillator, J. Phys. A, 35 (2002), L679–L686. doi: 10.1088/0305-4470/35/47/101.  Google Scholar

[9]

G. Chavchanidze, Non-Noether symmetries and their influence on phase space geometry, J. Geom. Phys., 48 (2003), 190-202.  doi: 10.1016/S0393-0440(03)00040-8.  Google Scholar

[10]

G. Chavchanidze, Non-Noether symmetries in Hamiltonian dynamical systems, Mem. Diff. Eqs. Math. Phys., 36 (2005), 81-134.   Google Scholar

[11]

M. Crampin, Constants of the motion in Lagrangian mechanics, Int. J. Theor. Phys., 16 (1977), 741-754.  doi: 10.1007/BF01807231.  Google Scholar

[12]

M. Crampin, A note on non-Noether constants of motion, Phys. Lett. A, 95 (1983), 209-212.  doi: 10.1016/0375-9601(83)90605-9.  Google Scholar

[13]

M. Crampin, W. Sarlet and G. Thompson, Bi-differential calculi and bi-Hamiltonian systems, J. Phys. A, 33 (2000), L177–180. doi: 10.1088/0305-4470/33/20/101.  Google Scholar

[14]

A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Reduction of presymplectic manifolds with symmetry, Rev. Math. Phys., 11 (1999), 1209-1247.  doi: 10.1142/S0129055X99000386.  Google Scholar

[15]

A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector field formulation of Hamiltonian field theories: Equations and symmetries, J. Phys. A: Math. Gen., 32 (1999), 8461-8484.  doi: 10.1088/0305-4470/32/48/309.  Google Scholar

[16]

G. Falqui, F. Magri and M. Pedroni, Bihamiltonian geometry and separation of variables for Toda lattices, J. Nonlinear Math. Phys., 8 (2001), suppl., 118–127. doi: 10.2991/jnmp.2001.8.s.21.  Google Scholar

[17]

J. Gaset, P. D. Prieto-Martínez and N. Román-Roy, Variational principles and symmetries on fibered multisymplectic manifolds, Comm. Math., 24 (2016), 137-152.  doi: 10.1515/cm-2016-0010.  Google Scholar

[18]

B. Jovanović, Noether symmetries and integrability in Hamiltonian time-dependent mechanics, Theor. App. Mechanics, 43 (2016), 255-273.  doi: 10.2298/TAM160121009J.  Google Scholar

[19]

P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and its Applications, 35. D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.  Google Scholar

[20]

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

[21]

F. A. Lunev, Analog of Noether's theorem for non-Noether and nonlocal symmetries, Theor. Math. Phys., 84 (1990), 816-820.  doi: 10.1007/BF01017679.  Google Scholar

[22]

M. Lutzky, Origin of non-Noether invariants, Phys. Lett. A, 75 (1980), 8-10.  doi: 10.1016/0375-9601(79)90258-5.  Google Scholar

[23]

M. Lutzky, New classes of conserved quantities associated with non-Noether symmetries, J. Phys. A, 15 (1982), L87–L91. doi: 10.1088/0305-4470/15/3/001.  Google Scholar

[24]

C. M. Marle and J. Nunes da Costa, Master symmetries and bi-Hamiltonian structures for the relativistic Toda lattice, J. Phys. A, 30 (1997), 7551-7556.  doi: 10.1088/0305-4470/30/21/025.  Google Scholar

[25]

G. Marmo and N. Mukunda, Symmetries and constants of the motion in the Lagrangian formalism on $TQ$: Beyond point transformations, Nuovo Cim B, 92 (1986), 1-12.  doi: 10.1007/BF02729691.  Google Scholar

[26]

G. Marmo, E. J. Saletan, A. Simoni and B. Vitale, Dynamical Systems, a Differential Geometric Approach to Symmetry and Reduction, John Wiley & Sons, Ltd., Chichester, 1985.  Google Scholar

[27]

J. C. Marrero, N. Román-Roy, M. Salgado and S. Vilariño, Reduction of polysymplectic manifolds, J. Phys. A, 48 (2015), 055206, 43 pp. doi: 10.1088/1751-8113/48/5/055206.  Google Scholar

[28]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Second edition, Texts in Applied Mathematics, 17. Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[29]

J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5 (1974), 121-130.  doi: 10.1016/0034-4877(74)90021-4.  Google Scholar

[30]

J. Mateos-Guilarte and M. S. Plyushchay, Perfectly invisible $\mathcal{PT}$-symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry, J. High Energy Phys., 2017 (2017), 061, front matter+35 pp. doi: 10.1007/JHEP12(2017)061.  Google Scholar

[31]

M. F. Rañada, Integrable three-particle systems, hidden symmetries and deformation of the Calogero-Moser system, J. Math. Phys., 36 (1995), 3541-3558.  doi: 10.1063/1.530980.  Google Scholar

[32]

M. F. Rañada, Superintegrable $n = 2$ systems, quadratic constants of motion, and potential of Drach, J. Math. Phys., 38 (1997), 4165-4178.  doi: 10.1063/1.532089.  Google Scholar

[33]

M. F. Rañada, Dynamical symmetries, bi-Hamiltonian structures and superintegrable $n = 2$ systems, J. Math. Phys., 41 (2000), 2121-2134.  doi: 10.1063/1.533230.  Google Scholar

[34]

N. Román-Roy, M. Salgado and S. Vilariño, Higher-order Noether symmetries in $k$-symplectic Hamiltonian field theory, Int. J. Geom. Methods Mod. Phys., 10 (2013), 1360013, 9 pp. doi: 10.1142/S021988781360013X.  Google Scholar

[35]

G. Rosensteel and J. P. Draayer, Symmetry algebra of the anisotropic harmonic oscillator with commensurate frequencies, J. Phys. A, 22 (1989), 1323-1327.  doi: 10.1088/0305-4470/22/9/021.  Google Scholar

[36]

W. Sarlet and F. Cantrijn, Higher-order Noether symmetries and constants of the motion, J. Phys. A, 14 (1981), 479-492.  doi: 10.1088/0305-4470/14/2/023.  Google Scholar

[37]

W. Sarlet and F. Cantrijn, Generalizations of Noether's theorem in classical mechanics, SIAM Rev., 23 (1981), 467-494.  doi: 10.1137/1023098.  Google Scholar

[38]

Yu. B. Suris, On the bi-Hamiltonian structure of Toda and relativistic Toda lattices, Phys. Lett. A, 180 (1993), 419-429.  doi: 10.1016/0375-9601(93)90293-9.  Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, Second edition, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. doi: 10.1090/chel/364.  Google Scholar

[2]

A. Arancibia and M. S. Plyushchay, Chiral asymmetry in propagation of soliton defects in crystalline backgrounds, Phys. Rev. D, 92 (2015), 105009, 20 pp. doi: 10.1103/PhysRevD.92.105009.  Google Scholar

[3]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[4]

P. Birtea and R. M. Tudoran, Non-Noether conservation laws, Int. J. Geom. Methods Mod. Phys., 9 (2012), 1220004, 5 pp. doi: 10.1142/S0219887812200046.  Google Scholar

[5]

A. V. Bolsinov, Compatible Poisson brackets on Lie algebras and the completeness of families of functions in involution, Math. USSR-Izvestiya, 38 (1992), 69-90.  doi: 10.1070/IM1992v038n01ABEH002187.  Google Scholar

[6]

A. V. Bolsinov and A. V. Borisov, Compatible Poisson brackets on Lie algebras, Math. Notes, 72 (2002), 10-30.  doi: 10.1023/A:1019856702638.  Google Scholar

[7]

J. F. Cariñena and L. A. Ibort, Non-Noether constants of motion, J. Phys. A, 16 (1983), 1-7.  doi: 10.1088/0305-4470/16/1/010.  Google Scholar

[8]

J. F. Cariñena, G. Marmo and M. F. Rañada, Non-symplectic symmetries and bi-Hamiltonian structures of the rational harmonic oscillator, J. Phys. A, 35 (2002), L679–L686. doi: 10.1088/0305-4470/35/47/101.  Google Scholar

[9]

G. Chavchanidze, Non-Noether symmetries and their influence on phase space geometry, J. Geom. Phys., 48 (2003), 190-202.  doi: 10.1016/S0393-0440(03)00040-8.  Google Scholar

[10]

G. Chavchanidze, Non-Noether symmetries in Hamiltonian dynamical systems, Mem. Diff. Eqs. Math. Phys., 36 (2005), 81-134.   Google Scholar

[11]

M. Crampin, Constants of the motion in Lagrangian mechanics, Int. J. Theor. Phys., 16 (1977), 741-754.  doi: 10.1007/BF01807231.  Google Scholar

[12]

M. Crampin, A note on non-Noether constants of motion, Phys. Lett. A, 95 (1983), 209-212.  doi: 10.1016/0375-9601(83)90605-9.  Google Scholar

[13]

M. Crampin, W. Sarlet and G. Thompson, Bi-differential calculi and bi-Hamiltonian systems, J. Phys. A, 33 (2000), L177–180. doi: 10.1088/0305-4470/33/20/101.  Google Scholar

[14]

A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Reduction of presymplectic manifolds with symmetry, Rev. Math. Phys., 11 (1999), 1209-1247.  doi: 10.1142/S0129055X99000386.  Google Scholar

[15]

A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector field formulation of Hamiltonian field theories: Equations and symmetries, J. Phys. A: Math. Gen., 32 (1999), 8461-8484.  doi: 10.1088/0305-4470/32/48/309.  Google Scholar

[16]

G. Falqui, F. Magri and M. Pedroni, Bihamiltonian geometry and separation of variables for Toda lattices, J. Nonlinear Math. Phys., 8 (2001), suppl., 118–127. doi: 10.2991/jnmp.2001.8.s.21.  Google Scholar

[17]

J. Gaset, P. D. Prieto-Martínez and N. Román-Roy, Variational principles and symmetries on fibered multisymplectic manifolds, Comm. Math., 24 (2016), 137-152.  doi: 10.1515/cm-2016-0010.  Google Scholar

[18]

B. Jovanović, Noether symmetries and integrability in Hamiltonian time-dependent mechanics, Theor. App. Mechanics, 43 (2016), 255-273.  doi: 10.2298/TAM160121009J.  Google Scholar

[19]

P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and its Applications, 35. D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.  Google Scholar

[20]

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

[21]

F. A. Lunev, Analog of Noether's theorem for non-Noether and nonlocal symmetries, Theor. Math. Phys., 84 (1990), 816-820.  doi: 10.1007/BF01017679.  Google Scholar

[22]

M. Lutzky, Origin of non-Noether invariants, Phys. Lett. A, 75 (1980), 8-10.  doi: 10.1016/0375-9601(79)90258-5.  Google Scholar

[23]

M. Lutzky, New classes of conserved quantities associated with non-Noether symmetries, J. Phys. A, 15 (1982), L87–L91. doi: 10.1088/0305-4470/15/3/001.  Google Scholar

[24]

C. M. Marle and J. Nunes da Costa, Master symmetries and bi-Hamiltonian structures for the relativistic Toda lattice, J. Phys. A, 30 (1997), 7551-7556.  doi: 10.1088/0305-4470/30/21/025.  Google Scholar

[25]

G. Marmo and N. Mukunda, Symmetries and constants of the motion in the Lagrangian formalism on $TQ$: Beyond point transformations, Nuovo Cim B, 92 (1986), 1-12.  doi: 10.1007/BF02729691.  Google Scholar

[26]

G. Marmo, E. J. Saletan, A. Simoni and B. Vitale, Dynamical Systems, a Differential Geometric Approach to Symmetry and Reduction, John Wiley & Sons, Ltd., Chichester, 1985.  Google Scholar

[27]

J. C. Marrero, N. Román-Roy, M. Salgado and S. Vilariño, Reduction of polysymplectic manifolds, J. Phys. A, 48 (2015), 055206, 43 pp. doi: 10.1088/1751-8113/48/5/055206.  Google Scholar

[28]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Second edition, Texts in Applied Mathematics, 17. Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[29]

J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5 (1974), 121-130.  doi: 10.1016/0034-4877(74)90021-4.  Google Scholar

[30]

J. Mateos-Guilarte and M. S. Plyushchay, Perfectly invisible $\mathcal{PT}$-symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry, J. High Energy Phys., 2017 (2017), 061, front matter+35 pp. doi: 10.1007/JHEP12(2017)061.  Google Scholar

[31]

M. F. Rañada, Integrable three-particle systems, hidden symmetries and deformation of the Calogero-Moser system, J. Math. Phys., 36 (1995), 3541-3558.  doi: 10.1063/1.530980.  Google Scholar

[32]

M. F. Rañada, Superintegrable $n = 2$ systems, quadratic constants of motion, and potential of Drach, J. Math. Phys., 38 (1997), 4165-4178.  doi: 10.1063/1.532089.  Google Scholar

[33]

M. F. Rañada, Dynamical symmetries, bi-Hamiltonian structures and superintegrable $n = 2$ systems, J. Math. Phys., 41 (2000), 2121-2134.  doi: 10.1063/1.533230.  Google Scholar

[34]

N. Román-Roy, M. Salgado and S. Vilariño, Higher-order Noether symmetries in $k$-symplectic Hamiltonian field theory, Int. J. Geom. Methods Mod. Phys., 10 (2013), 1360013, 9 pp. doi: 10.1142/S021988781360013X.  Google Scholar

[35]

G. Rosensteel and J. P. Draayer, Symmetry algebra of the anisotropic harmonic oscillator with commensurate frequencies, J. Phys. A, 22 (1989), 1323-1327.  doi: 10.1088/0305-4470/22/9/021.  Google Scholar

[36]

W. Sarlet and F. Cantrijn, Higher-order Noether symmetries and constants of the motion, J. Phys. A, 14 (1981), 479-492.  doi: 10.1088/0305-4470/14/2/023.  Google Scholar

[37]

W. Sarlet and F. Cantrijn, Generalizations of Noether's theorem in classical mechanics, SIAM Rev., 23 (1981), 467-494.  doi: 10.1137/1023098.  Google Scholar

[38]

Yu. B. Suris, On the bi-Hamiltonian structure of Toda and relativistic Toda lattices, Phys. Lett. A, 180 (1993), 419-429.  doi: 10.1016/0375-9601(93)90293-9.  Google Scholar

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