Reductions of topologically massive gravity. I: Hamiltonian analysis of second order degenerate Lagrangians. (English) Zbl 1383.83108
Summary: We present Skinner-Rusk and Hamiltonian formalisms of second order degenerate Clément and Sarıoğlu-Tekin Lagrangians. The Dirac-Bergmann constraint algorithm is employed to obtain Hamiltonian realizations of Lagrangian theories. The Gotay-Nester-Hinds algorithm [M. J. Gotay et al., ibid. 19, 2388–2399 (1978; Zbl 0418.58010)] is used to investigate Skinner-Rusk formalisms [R. Skinner and R. Rusk, ibid. 24, 2589–2594 (1983; Zbl 0556.70012); R. Skinner and R. Rusk, J. Math. Phys. 24, 2595–2601 (1983; Zbl 0556.70013)] of these systems.
{
©2018 American Institute of Physics}
©2018 American Institute of Physics}
MSC:
| 83D05 | Relativistic gravitational theories other than Einstein’s, including asymmetric field theories |
| 70S05 | Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems |
Keywords:
topologically massive gravity; Hamiltonian formalisms; Dirac-Bergmann constraint; Gotay-Nester-Hinds algorithm; Skinner-Rusk formalismsReferences:
| [1] | Clément, G., Classical Quantum Gravity, 9, 2615 (1992) · Zbl 0770.53046 · doi:10.1088/0264-9381/9/12/006 |
| [2] | Clément, G., Phys. Rev. D, 49, 5131 (1994) · doi:10.1103/physrevd.49.5131 |
| [3] | Clément, G., Classical Quantum Gravity, 11, L115 (1994) · doi:10.1088/0264-9381/11/9/001 |
| [4] | Sarıoğlu, Ö.; Tekin, B., Classical Quantum Gravity, 23, 7541 (2006) · Zbl 1107.83052 · doi:10.1088/0264-9381/23/24/023 |
| [5] | Deser, S.; Jackiw, R.; Templeton, S., Ann. Phys., 140, 372 (1982) · doi:10.1016/0003-4916(82)90164-6 |
| [6] | Deser, S.; Jackiw, R.; Templeton, S., Phys. Rev. Lett., 48, 975 (1982) · doi:10.1103/physrevlett.48.975 |
| [7] | Bolonek, K.; Kosiński, P., Hamiltonian structures for pais-uhlenbeck oscillator, Acta Phys. Pol., B, 36, 2115 (2005) |
| [8] | Mannheim, P. D.; Davidson, A., Phys. Rev. A, 71, 042110 (2005) · Zbl 1227.81222 · doi:10.1103/physreva.71.042110 |
| [9] | Masterov, I., Nucl. Phys. B, 902, 95 (2016) · Zbl 1332.81059 · doi:10.1016/j.nuclphysb.2015.11.011 |
| [10] | Masterov, I., Nucl. Phys. B, 907, 495 (2016) · Zbl 1336.81032 · doi:10.1016/j.nuclphysb.2016.04.025 |
| [11] | Mostafazadeh, A., Phys. Lett. A, 375, 93 (2010) · Zbl 1241.70023 · doi:10.1016/j.physleta.2010.10.050 |
| [12] | Batlee, C.; Gomis, J.; Pons, J. M.; Roman-Roy, N., J. Phys. A: Math. Gen., 21, 2693 (1988) · Zbl 0658.70017 · doi:10.1088/0305-4470/21/12/013 |
| [13] | Crampin, M.; Sarlet, W.; Cantrijn, F., Math. Proc. Cambridge Philos. Soc., 99, 565 (1986) · Zbl 0609.58049 · doi:10.1017/s0305004100064501 |
| [14] | Govaerts, J.; Rashid, M. S. (1994) |
| [15] | Gracia, X.; Pons, J. M.; Román-Roy, N., J. Math. Phys., 32, 2744 (1991) · Zbl 0778.58025 · doi:10.1063/1.529066 |
| [16] | Krupková, O., J. Math. Phys., 35, 6557 (1994) · Zbl 0823.70016 · doi:10.1063/1.530691 |
| [17] | Nakamura, T.; Hamamoto, S., Prog. Theor. Phys., 95, 469 (1996) · doi:10.1143/ptp.95.469 |
| [18] | Nesterenko, V. V., J. Phys. A: Math. Gen., 22, 1673 (1989) · Zbl 0695.58014 · doi:10.1088/0305-4470/22/10/021 |
| [19] | Pons, J. M., Lett. Math. Phys., 17, 181 (1989) · Zbl 0688.70010 · doi:10.1007/bf00401583 |
| [20] | Rashid, M. S.; Khalil, S. S., Int. J. Mod. Phys. A, 11, 4551 (1996) · Zbl 1044.70505 · doi:10.1142/s0217751x96002108 |
| [21] | Bergmann, P. G., Helv. Phys. Acta, Suppl., 4, 79 (1956) |
| [22] | Dirac, P. A. M., Lectures on Quantum Mechanics (1964) · Zbl 0141.44603 |
| [23] | Dirac, P. A. M., Proc. R. Soc. London, Ser. A, 246, 326 (1958) · Zbl 0080.41403 · doi:10.1098/rspa.1958.0142 |
| [24] | Sudarshan, E. C. G.; Mukunda, N., Classical Dynamics: A Modern Perspective (1974) · Zbl 0329.70001 |
| [25] | Gotay, M. J.; Nester, J. M., Ann. l��IHP Phys. Théo, 30, 129 (1979) · Zbl 0414.58015 |
| [26] | Gotay, M. J.; Nester, J. M., Lect. Notes Math., 775, 78 (1980) · doi:10.1007/bfb0092022 |
| [27] | Gotay, M. J.; Nester, J. M., J. Phys. A: Math. Gen., 17, 3063 (1984) · doi:10.1088/0305-4470/17/15/023 |
| [28] | Gotay, M. J.; Nester, J. M.; Hinds, G., J. Math. Phys., 19, 2388 (1978) · Zbl 0418.58010 · doi:10.1063/1.523597 |
| [29] | Ostrogradsky, M., Mem. Ac. St. Petersbourg VI, 4, 385 (1850) |
| [30] | Colombo, L.; Prieto-Martínez, P. D., J. Math. Phys., 57, 082901 (2016) · Zbl 1362.37118 · doi:10.1063/1.4960822 |
| [31] | Chen, T. J.; Fasiello, M.; Lim, E. A.; Tolley, A. J., J. Cosmol. Astropart. Phys., 2013, 042 · doi:10.1088/1475-7516/2013/02/042 |
| [32] | de León, M.; Marín-Solano, J.; Marrero, J. C.; Muñoz-Lecanda, M. C.; Román-Roy, N., Fortschr. Phys., 50, 105 (2002) · Zbl 1010.37031 · doi:10.1002/1521-3978(200203)50:2<105::aid-prop105>3.0.co;2-n |
| [33] | Cruz, M.; Gómez-Cortés, R.; Molgado, A.; Rojas, E., J. Math. Phys., 57, 062903 (2016) · Zbl 1347.70029 · doi:10.1063/1.4954804 |
| [34] | Skinner, R., J. Math. Phys., 24, 2581 (1983) · Zbl 0532.70009 · doi:10.1063/1.525653 |
| [35] | Skinner, R.; Rusk, R., J. Math. Phys., 24, 2595 (1983) · Zbl 0556.70013 · doi:10.1063/1.525655 |
| [36] | Prieto-Martínez, P. D.; Román-Roy, N., J. Phys. A: Math. Theor., 44, 385203 (2011) · Zbl 1242.70031 · doi:10.1088/1751-8113/44/38/385203 |
| [37] | Campos, C. M.; de León, M.; De Diego, D. M.; Vankerschaver, J., J. Phys. A: Math. Theor., 42, 475207 (2009) · Zbl 1231.58005 · doi:10.1088/1751-8113/42/47/475207 |
| [38] | Vitagliano, L., J. Geom. Phys., 60, 857 (2010) · Zbl 1208.70027 · doi:10.1016/j.geomphys.2010.02.003 |
| [39] | Colombo, L.; de Diego, D. M., J. Geom. Mech., 6, 451 (2014) · Zbl 1307.70018 · doi:10.3934/jgm.2014.6.451 |
| [40] | Barbero-Liñán, M.; Echeverría-Enrí quez, A.; De Diego, D. M.; Muñoz-Lecanda, M. C.; Román-Roy, N., J. Phys. A: Math. Theor., 40, 12071 (2007) · Zbl 1126.70011 · doi:10.1088/1751-8113/40/40/005 |
| [41] | Abraham, R.; Marsden, J. E., Foundations of Mechanics (1978) · Zbl 0393.70001 |
| [42] | De León, M.; Rodrigues, P. R., Generalized Classical Mechanics and Field Theory: A Geometrical Approach of Lagrangian and Hamiltonian Formalisms Involving Higher Order Derivatives (2011) |
| [43] | Suri, A., Mediterr. J. Math., 14, 5 (2016) · Zbl 1364.58003 · doi:10.1007/s00009-016-0812-7 |
| [44] | Rund, H., Ann. Mat. Pura Appl., 64, 99 (1964) · Zbl 0124.05603 · doi:10.1007/BF02410049 |
| [45] | Dunin-Barkowski, P. I.; Sleptsov, A. V., Theor. Math. Phys., 158, 61 (2009) · Zbl 1241.70047 · doi:10.1007/s11232-009-0005-7 |
| [46] | Urban, Z.; Krupka, D., Publ. Math. Debrecen, 82, 59 (2013) · Zbl 1299.34036 · doi:10.5486/pmd.2013.5265 |
| [47] | Zermelo, E., Untersuchungen zur Variations-Rechnung (1894) · JFM 25.0649.02 |
| [48] | Kondo, K. (1991) |
| [49] | Saito, Y.; Sugano, R.; Ohta, T.; Kimura, T., J. Math. Phys., 30, 1122 (1989) · Zbl 0682.70015 · doi:10.1063/1.528331 |
| [50] | Prieto-Martínez, P. D.; Román-Roy, N., J. Math. Phys., 53, 032901 (2012) · Zbl 1274.37033 · doi:10.1063/1.3692326 |
| [51] | Schmidt, H. J., Phys. Rev. D, 49, 6354 (1994) · doi:10.1103/physrevd.49.6354 |
| [52] | Schmidt, H. J. (1995) |
| [53] | Esen, O.; Guha, P. (2016) |
| [54] | Andrzejewski, K.; Gonera, J.; Machalski, P.; Maślanka, P., Phys. Rev. D, 82, 045008 (2010) · doi:10.1103/physrevd.82.045008 |
| [55] | Andrzejewski, K.; Gonera, J.; Maslanka, P. (2007) |
| [56] | Gay-Balmaz, F.; Holm, D. D.; Ratiu, T. S., Bull. Braz. Math. Soc., New Ser., 42, 579 (2011) · Zbl 1319.70022 · doi:10.1007/s00574-011-0030-7 |
| [57] | Cendra, H.; Marsden, J. E.; Ratiu, T. S., Lagrangian Reduction by Stages (2001) · Zbl 1193.37072 · doi:10.1090/memo/0722 |
| [58] | Marsden, J. E.; Ratiu, T., Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems (1998) |
| [59] | Colombo, L.; de León, M.; Prieto-Martínez, P.; Román-Roy, N., Int. J. Geom. Methods Mod. Phys., 11, 1460037 (2014) · Zbl 1305.70039 · doi:10.1142/s0219887814600378 |
| [60] | de Leon, M.; Pitanga, P.; Rodrigues, P. R., J. Math. Phys., 35, 6546 (1994) · Zbl 0824.70012 · doi:10.1063/1.530890 |
| [61] | Derrick, G. H., J. Math. Phys., 28, 64 (1987) · Zbl 0618.70008 · doi:10.1063/1.527809 |
| [62] | Gerdt, V.; Khvedelidze, A.; Palii, Y., 135 (2006) |
| [63] | Kasper, U., Gen. Relativ. Gravitation, 29, 221 (1997) · Zbl 0882.58064 · doi:10.1023/a:1010292128733 |
| [64] | Lukierski, J.; Stichel, P. C.; Zakrzewski, W. J., Ann. Phys., 260, 224 (1997) · Zbl 0974.37041 · doi:10.1006/aphy.1997.5729 |
| [65] | Shadwick, W. F., Lett. Math. Phys., 6, 409 (1982) · Zbl 0514.58013 · doi:10.1007/bf00405859 |
| [66] | Miron, R., The Geometry of Higher-Order Lagrange Spaces Applications to Mechanics and Physics (1997) · Zbl 0877.53001 |
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.
