Continuous singularities in hamiltonian relative equilibria with abelian momentum isotropy
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
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
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
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
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
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
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
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
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
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
G. Chavchanidze, Non-Noether symmetries in Hamiltonian dynamical systems, Mem. Diff. Eqs. Math. Phys., 36 (2005), 81-134. Google Scholar
M. Crampin, Constants of the motion in Lagrangian mechanics, Int. J. Theor. Phys., 16 (1977), 741-754. doi: 10.1007/BF01807231. Google Scholar
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
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
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
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
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
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
B. Jovanović, Noether symmetries and integrability in Hamiltonian time-dependent mechanics, Theor. App. Mechanics, 43 (2016), 255-273. doi: 10.2298/TAM160121009J. Google Scholar
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
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
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
M. Lutzky, Origin of non-Noether invariants, Phys. Lett. A, 75 (1980), 8-10. doi: 10.1016/0375-9601(79)90258-5. Google Scholar
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024
Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030
Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020406
Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2020 doi: 10.3934/jgm.2021001
João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138
Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020368
Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176
Simone Fiori. Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport. Mathematical Control & Related Fields, 2021, 11 (1) : 143-167. doi: 10.3934/mcrf.2020031
Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 0: 331-348. doi: 10.3934/jmd.2020012
Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2020 doi: 10.3934/jgm.2020031
Harrison Bray. Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds. Journal of Modern Dynamics, 2020, 16: 305-329. doi: 10.3934/jmd.2020011
Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020377
Philippe G. Lefloch, Cristinel Mardare, Sorin Mardare. Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 341-365. doi: 10.3934/dcds.2009.23.341
Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020409
Lin Shi, Dingshi Li, Kening Lu. Limiting behavior of unstable manifolds for spdes in varying phase spaces. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021020