Symmetries, conservation and dissipation in time-dependent contact systems. (English) Zbl 07771673
Summary: In contact Hamiltonian systems, the so-called dissipated quantities are akin to conserved quantities in classical Hamiltonian systems. In this article, a Noether’s theorem for non-autonomous contact Hamiltonian systems is proved, characterizing a class of symmetries which are in bijection with dissipated quantities. Other classes of symmetries which preserve (up to a conformal factor) additional structures, such as the contact form or the Hamiltonian function, are also studied. Furthermore, making use of the geometric structures of the extended tangent bundle, additional classes of symmetries for time-dependent contact Lagrangian systems are introduced. The results are illustrated with several examples. In particular, the two-body problem with time-dependent friction is presented, which could be interesting in celestial mechanics.
© 2023 The Authors. Fortschritte der Physik published by Wiley-VCH GmbH
© 2023 The Authors. Fortschritte der Physik published by Wiley-VCH GmbH
MSC:
| 37J55 | Contact systems |
| 70H33 | Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics |
| 70F05 | Two-body problems |
References:
| [1] | A.Bravetti, Entropy2017, 19, 535. |
| [2] | A.Bravetti, H.Cruz, D.Tapias, Ann. Phys.2017, 376, 17. · Zbl 1364.37138 |
| [3] | M.deLeón, M.Lainz, Revista de la Real Academia de Ciencias Canaria, 2019, pp. 1-46. |
| [4] | M.Lainz, Contact Hamiltonian Systems. 2022. |
| [5] | M.deLeón, M.Lainz Valcázar, J. Math. Phys2019, 60, 102902. · Zbl 1427.70039 |
| [6] | J.Gaset, X.Grácia, M. C.Muñoz‐Lecanda, X.Rivas, N.Román‐Roy, Int. J. Geom. Methods Mod. Phys.2020, 17, 2050090. · Zbl 07809134 |
| [7] | X.Rivas, Geometrical aspects of contact mechanical systems and field theories. 2022. |
| [8] | J.Gaset, in: Extended Abstracts GEOMVAP 2019, Trends in Mathematics (Eds M.Alberich‐Carramiñana (ed.), G.Blanco (ed.), I. GálvezCarrillo (ed.), M.Garrote‐López (ed.), E.Miranda (ed.)), Springer International Publishing, 2019, p. 71. |
| [9] | A. A.Simoes, M.deLeón, M. L.Valcázar, D. M.Martín de Diego, Proc. Math. Phys. Eng. Sci2020, 476, 20200244. · Zbl 1472.53109 |
| [10] | A.Bravetti, Int. J. Geom. Methods Mod. Phys.2019, 16, 1940003. · Zbl 1421.80002 |
| [11] | D.Eberard, B. M.Maschke, A. J.van derSchaft, Reports on Mathematical Physics.2007, 60, 175. · Zbl 1210.80001 |
| [12] | F.Gay‐Balmaz, H.Yoshimura, Entropy2018, 21, 8. |
| [13] | R.Mrugała, in: Thermodynamics of Energy Conversion and Transport, (Eds S.Sieniutycz (ed.), A.De Vos (ed.)), Springer, New York pp. 257-285. |
| [14] | F. M.Ciaglia, H.Cruz, G.Marmo, Ann. Phys.2018, 398, 159. https://doi.org/10.1016/j.aop.2018.09.012. · Zbl 1404.81146 · doi:10.1016/j.aop.2018.09.012 |
| [15] | M.deLeón, V. M.Jiménez, M.Lainz, J. Geom. Mech.2021, 13, 25. · Zbl 1496.37071 |
| [16] | J.Gaset, A.Marín‐Salvador, Int. J. Geom. Methods Mod. Phys.2022, 19, 2250156. · Zbl 07850397 |
| [17] | J.Gaset, A.Mas, J. Geom. Mech.2023, 15, 357. · Zbl 1523.70036 |
| [18] | J.deLucas, X.Rivas, Contact Lie systems: Theory and applications. arXiv: https://arxiv.org/abs/2207.040382207.04038. 2022. |
| [19] | M.deLeón, M.Lainz, M. C.Muñoz‐Lecanda, J. Nonlinear Sci.2023, 33, 9. issn: 0938‐8974, 1432‐1467 |
| [20] | J.Gaset, X.Grácia, M. C.Muñoz‐Lecanda, X.Rivas, N.Román‐Roy, Ann. Phys.2020, 414, 168092. · Zbl 1433.53042 |
| [21] | J.Gaset, X.Grácia, M. C.Muñoz‐Lecanda, X.Rivas, N.Román‐Roy, Rep. Math. Phys.2021, 87, 347. · Zbl 1487.70095 |
| [22] | M.deLeón, C.Sardón, J. Phys. A: Math. Theor.2017, 50, 255205. · Zbl 1425.70031 |
| [23] | M.deLeón, J.Gaset, X.Grácia, M. C.Muñoz‐Lecanda, X.Rivas, Mon Hefte Math.2022, 1436. |
| [24] | X.Rivas, D.Torres, J. Geom. Mech.2022, 16, 1. |
| [25] | E.Noether, Transport Theory and Statistical Physics1971, 1, 186. · Zbl 0292.49008 |
| [26] | Y.Ne’eman, in: The Heritage of Emmy Noether (Ramat‐Gan, 1996), Vol. 12 of Israel Math. Conf. Proc., pp. 83-101. Bar‐Ilan Univ., Ramat Gan. · Zbl 0928.01013 |
| [27] | Y.Kosmann‐Schwarzbach, The Noether Theorems. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York. · Zbl 0981.53080 |
| [28] | J. F.Carinena, C.Lopez, E.Martinez, J. Phys. A: Math. Gen.1989, 22, 4777. · Zbl 0739.58016 |
| [29] | J. F.Carinena, E.Martinez, J. Phys. A: Math. Gen.1989, 22, 2659. · Zbl 0709.58015 |
| [30] | J. F.Cariñena, H.Figueroa, Rep. Math. Phys.1994, 34, 277. · Zbl 0846.58008 |
| [31] | M.deLeón, P. R.Rodrigues, Methods of Differential Geometry in Analytical Mechanics. Number 158 in North‐Holland Mathematics Studies. North‐Holland. |
| [32] | M.deLeón, D.Martín de Diego, Extracta Mathematicae1994, 9, 32. · Zbl 1027.37503 |
| [33] | M.deLeón, D.Martín de Diego, Internat. J. Theoret. Phys.1996, 35, 975. · Zbl 0856.58034 |
| [34] | F. A.Lunev, Teoret. Mat. Fiz.1990, 84, 205. |
| [35] | D. S.Djukic, B. D.Vujanovic, Acta Mechanica.1975, 23, 17. · Zbl 0328.70013 |
| [36] | C.Ferrario, A.Passerini, J. Phys. A1990, 23, 5061. |
| [37] | G.Marmo, N.Mukunda, Nuov Cim B1986, 92, 1. |
| [38] | D. N. K.Marwat, A. H.Kara, F. M.Mahomed, Int. J. Theoret. Phys.2007, 46, 3022. · Zbl 1133.70008 |
| [39] | G.Prince, Bull. Aust. Math. Soc.1983, 27, 53. · Zbl 0495.58014 |
| [40] | G.Prince, Bull. Aust. Math. Soc.1985, 32, 299. · Zbl 0578.58018 |
| [41] | W.Sarlet, F.Cantrijn, SIAM Rev.1981, 23, 467494. |
| [42] | W.Sarlet, J. Phys. A: Math. Gen.1983, 15, L229. |
| [43] | A. J.van derSchaft, Internat. J. Theoret. Phys.2007, 24, 2095. |
| [44] | M.deLeón, M.Lainz, A.López‐Gordón, J. Math. Phys.2021, 62, 042901. · Zbl 1481.70067 |
| [45] | M.deLeón, M.Lainz Valcázar, J. Geom. Phys.2020, 153, 103651. · Zbl 1454.37061 |
| [46] | M.deLeón, M.Lainz, A.Mu niz‐Brea, Mathematics1993, 9, 1993. |
| [47] | M.deLeón, M.Lainz, A.López‐Gordón, X.Rivas, J. Geom. Phys.2023, 187, 104787. · Zbl 1514.37086 |
| [48] | R.Azuaje, A.Escobar‐Ruiz, J. Math. Phys.2023, 64, 033501. · Zbl 1511.53077 |
| [49] | R.Azuaje, Lie integrability for time‐independent and time‐dependent Hamiltonian systems. 2023. |
| [50] | A.Bravetti, A.Garcia‐Chung, J. Phys. A: Math. Theor2021, 54, 095205. · Zbl 1519.70019 |
| [51] | M.deLeón, J.Gaset, M.Lainz, Journal of Geometry and Physics2022, 176, 104500. · Zbl 1501.37067 |
| [52] | H.Geiges, An Introduction to Contact Topology. Cambridge Studies in Advanced Mathematics. Cambridge University Press. |
| [53] | P.Libermann, C. M.Marle, Symplectic Geometry and Analytical Mechanics, Springer Netherlands1987 · Zbl 0643.53002 |
| [54] | M.deLeón, M.Lainz Valcázar, Int. J. Geom. Methods Mod. Phys2019, 16, 1950158. · Zbl 07801950 |
| [55] | X.Grácia, Rep. Math. Phys.2000, 45, 67. · Zbl 0983.37078 |
| [56] | G.Herglotz, Berührungstransformationen. 1930. |
| [57] | N.Román‐Roy, J. Geom. Mech.1983, 12, 541. |
| [58] | C.López, E.Martínez, M. F.Rañada, J. Phys. A: Math. Gen.1999, 32, 1241. · Zbl 0939.70016 |
| [59] | K.Yano, S.Ishihara, Tangent and Cotangent Bundles; Differential Geometry, Marcel Dekker, Inc.1973. · Zbl 0262.53024 |
| [60] | X.Rivas, J. Math. Phys.2023, 64, 033507. · Zbl 1511.53083 |
| [61] | M.deLeón, J.Gaset, M. C.Muñoz‐Lecanda, X.Rivas, N.Román‐Roy, J. Phys. A: Math. Theor.2022, 56, 025201. |
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.
