A summary on symmetries and conserved quantities of autonomous Hamiltonian systems

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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 September 2020 Early access 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	G. Chavchanidze, Non-Noether symmetries in Hamiltonian dynamical systems, Mem. Diff. Eqs. Math. Phys., 36 (2005), 81-134.
[11]	M. Crampin, Constants of the motion in Lagrangian mechanics, Int. J. Theor. Phys., 16 (1977), 741-754. doi: 10.1007/BF01807231.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	B. Jovanović, Noether symmetries and integrability in Hamiltonian time-dependent mechanics, Theor. App. Mechanics, 43 (2016), 255-273. doi: 10.2298/TAM160121009J.
[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.
[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.
[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.
[22]	M. Lutzky, Origin of non-Noether invariants, Phys. Lett. A, 75 (1980), 8-10. doi: 10.1016/0375-9601(79)90258-5.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[37]	W. Sarlet and F. Cantrijn, Generalizations of Noether's theorem in classical mechanics, SIAM Rev., 23 (1981), 467-494. doi: 10.1137/1023098.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	G. Chavchanidze, Non-Noether symmetries in Hamiltonian dynamical systems, Mem. Diff. Eqs. Math. Phys., 36 (2005), 81-134.
[11]	M. Crampin, Constants of the motion in Lagrangian mechanics, Int. J. Theor. Phys., 16 (1977), 741-754. doi: 10.1007/BF01807231.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	B. Jovanović, Noether symmetries and integrability in Hamiltonian time-dependent mechanics, Theor. App. Mechanics, 43 (2016), 255-273. doi: 10.2298/TAM160121009J.
[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.
[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.
[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.
[22]	M. Lutzky, Origin of non-Noether invariants, Phys. Lett. A, 75 (1980), 8-10. doi: 10.1016/0375-9601(79)90258-5.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[37]	W. Sarlet and F. Cantrijn, Generalizations of Noether's theorem in classical mechanics, SIAM Rev., 23 (1981), 467-494. doi: 10.1137/1023098.
[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.

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