A summary on symmetries and conserved quantities of autonomous Hamiltonian systems. (English) Zbl 1460.37052
The existence of symmetries of Hamiltonian and Lagrangian systems is related to the existence of conserved quantities. As it is well known, the standard procedure to obtain conserved quantities consists in introducing the so-called Noether symmetries, and then use the Noether theorem. However, these kinds of symmetries do not exhaust the set of symmetries. As it is known, there are symmetries which are not of Noether type, but they also generate conserved quantities and they are sometimes called hidden symmetries. In this paper the author establishes a complete scheme of classification of all the different kinds of symmetries of Hamiltonian systems, explaining how to obtain the associated conserved quantities in each case. The author follows the same lines of argument as in the analysis made in [W. Sarlet and F. Cantrijn, J. Phys. A, Math. Gen. 14, 479–492 (1981; Zbl 0464.58010)] for nonautonomous Lagrangian systems, where the authors obtained conserved quantities for different kinds of symmetries that do not leave the Poincaré-Cartan form invariant.
Reviewer: Nicolai K. Smolentsev (Kemerovo)
MSC:
| 37J06 | General theory of finite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, invariants |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37J39 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.) |
| 70H33 | Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics |
| 53D05 | Symplectic manifolds (general theory) |
| 70S05 | Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems |
| 70S10 | Symmetries and conservation laws in mechanics of particles and systems |
Keywords:
symmetries; conserved quantities; Hamiltonian systems; Noether theorem; symplectic manifoldsCitations:
Zbl 0464.58010References:
| [1] | R. Abraham and J. E. Marsden, Foundations of Mechanics, Second edition, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. · Zbl 0393.70001 |
| [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. |
| [3] | V. I. Arnol’d, Mathematical Methods of Classical Mechanics, Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. · Zbl 0692.70003 |
| [4] | P. Birtea and R. M. Tudoran, Non-Noether conservation laws, Int. J. Geom. Methods Mod. Phys., 9 (2012), 1220004, 5 pp. · Zbl 1261.37022 |
| [5] | A. V. Bolsinov, Compatible Poisson brackets on Lie algebras and the completeness of families of functions in involution, Math. USSR-Izvestiya, 38, 69-90 (1992) · Zbl 0744.58030 · doi:10.1070/IM1992v038n01ABEH002187 |
| [6] | A. V. Bolsinov; A. V. Borisov, Compatible Poisson brackets on Lie algebras, Math. Notes, 72, 10-30 (2002) · Zbl 1042.37041 · doi:10.1023/A:1019856702638 |
| [7] | J. F. Cariñena; L. A. Ibort, Non-Noether constants of motion, J. Phys. A, 16, 1-7 (1983) · Zbl 0521.58055 · 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. · Zbl 1168.70316 |
| [9] | G. Chavchanidze, Non-Noether symmetries and their influence on phase space geometry, J. Geom. Phys., 48, 190-202 (2003) · Zbl 1056.70010 · doi:10.1016/S0393-0440(03)00040-8 |
| [10] | G. Chavchanidze, Non-Noether symmetries in Hamiltonian dynamical systems, Mem. Diff. Eqs. Math. Phys., 36, 81-134 (2005) · Zbl 1094.37035 |
| [11] | M. Crampin, Constants of the motion in Lagrangian mechanics, Int. J. Theor. Phys., 16, 741-754 (1977) · Zbl 0386.70004 · doi:10.1007/BF01807231 |
| [12] | M. Crampin, A note on non-Noether constants of motion, Phys. Lett. A, 95, 209-212 (1983) · 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. · Zbl 0989.37063 |
| [14] | A. Echeverría-Enríquez; M. C. Muñoz-Lecanda; N. Román-Roy, Reduction of presymplectic manifolds with symmetry, Rev. Math. Phys., 11, 1209-1247 (1999) · Zbl 0983.53056 · doi:10.1142/S0129055X99000386 |
| [15] | A. Echeverría-Enríquez; M. C. Muñoz-Lecanda; N. Román-Roy, Multivector field formulation of Hamiltonian field theories: Equations and symmetries, J. Phys. A: Math. Gen., 32, 8461-8484 (1999) · Zbl 0982.70019 · 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. · Zbl 0981.37018 |
| [17] | J. Gaset; P. D. Prieto-Martínez; N. Román-Roy, Variational principles and symmetries on fibered multisymplectic manifolds, Comm. Math., 24, 137-152 (2016) · Zbl 1406.49054 · doi:10.1515/cm-2016-0010 |
| [18] | B. Jovanović, Noether symmetries and integrability in Hamiltonian time-dependent mechanics, Theor. App. Mechanics, 43, 255-273 (2016) · Zbl 1474.37059 · 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. · Zbl 0643.53002 |
| [20] | C. López; E. Martínez; M. F. Rañada, Dynamical symmetries, non-Cartan symmetries and superintegrability of the \(n\)-dimensional harmonic oscillator, J. Phys. A, 32, 1241-1249 (1999) · Zbl 0939.70016 · 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, 816-820 (1990) · doi:10.1007/BF01017679 |
| [22] | M. Lutzky, Origin of non-Noether invariants, Phys. Lett. A, 75, 8-10 (1980) · 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. · Zbl 0522.70025 |
| [24] | C. M. Marle; J. Nunes da Costa, Master symmetries and bi-Hamiltonian structures for the relativistic Toda lattice, J. Phys. A, 30, 7551-7556 (1997) · Zbl 1004.37510 · doi:10.1088/0305-4470/30/21/025 |
| [25] | G. Marmo; N. Mukunda, Symmetries and constants of the motion in the Lagrangian formalism on \(TQ\): Beyond point transformations, Nuovo Cim B, 92, 1-12 (1986) · 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. · Zbl 0592.58031 |
| [27] | J. C. Marrero, N. Román-Roy, M. Salgado and S. Vilariño, Reduction of polysymplectic manifolds, J. Phys. A, 48 (2015), 055206, 43 pp. · Zbl 1321.53094 |
| [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. · Zbl 0933.70003 |
| [29] | J. Marsden; A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5, 121-130 (1974) · Zbl 0327.58005 · 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. · Zbl 1383.81220 |
| [31] | M. F. Rañada, Integrable three-particle systems, hidden symmetries and deformation of the Calogero-Moser system, J. Math. Phys., 36, 3541-3558 (1995) · Zbl 0844.58033 · 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, 4165-4178 (1997) · Zbl 0883.58014 · doi:10.1063/1.532089 |
| [33] | M. F. Rañada, Dynamical symmetries, bi-Hamiltonian structures and superintegrable \(n = 2\) systems, J. Math. Phys., 41, 2121-2134 (2000) · Zbl 1045.37034 · 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. · Zbl 1148.70013 |
| [35] | G. Rosensteel; J. P. Draayer, Symmetry algebra of the anisotropic harmonic oscillator with commensurate frequencies, J. Phys. A, 22, 1323-1327 (1989) · doi:10.1088/0305-4470/22/9/021 |
| [36] | W. Sarlet; F. Cantrijn, Higher-order Noether symmetries and constants of the motion, J. Phys. A, 14, 479-492 (1981) · Zbl 0464.58010 · doi:10.1088/0305-4470/14/2/023 |
| [37] | W. Sarlet; F. Cantrijn, Generalizations of Noether’s theorem in classical mechanics, SIAM Rev., 23, 467-494 (1981) · Zbl 0474.70014 · doi:10.1137/1023098 |
| [38] | Yu. B. Suris, On the bi-Hamiltonian structure of Toda and relativistic Toda lattices, Phys. Lett. A, 180, 419-429 (1993) · doi:10.1016/0375-9601(93)90293-9 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
