We derive the equations of motion of an action-dependent version of the Einstein-Hilbert Lagrangian as a specific instance of the Herglotz variational problem. Action-dependent Lagrangians lead to dissipative dynamics, which cannot be obtained with the standard method of Lagrangian field theory. First-order theories of this kind are relatively well understood, but examples of singular or higher-order action-dependent field theories are scarce. This work constitutes an example of such a theory. By casting the problem in clear geometric terms, we are able to obtain a Lorentz invariant set of equations, which contrasts with previous attempts.
Citation: Jordi Gaset, Arnau Mas. A variational derivation of the field equations of an action-dependent Einstein-Hilbert Lagrangian[J]. Journal of Geometric Mechanics, 2023, 15(1): 357-374. doi: 10.3934/jgm.2023014
| [1] | Xavier Rivas, Daniel Torres . Lagrangian–Hamiltonian formalism for cocontact systems. Journal of Geometric Mechanics, 2023, 15(1): 1-26. doi: 10.3934/jgm.2023001 |
| [2] | Elias Maciel, Inocencio Ortiz, Christian E. Schaerer . A Herglotz-based integrator for nonholonomic mechanical systems. Journal of Geometric Mechanics, 2023, 15(1): 287-318. doi: 10.3934/jgm.2023012 |
| [3] | Jacob R. Goodman . Local minimizers for variational obstacle avoidance on Riemannian manifolds. Journal of Geometric Mechanics, 2023, 15(1): 59-72. doi: 10.3934/jgm.2023003 |
| [4] | Valentin Duruisseaux, Melvin Leok . Time-adaptive Lagrangian variational integrators for accelerated optimization. Journal of Geometric Mechanics, 2023, 15(1): 224-255. doi: 10.3934/jgm.2023010 |
| [5] | Robert I McLachlan, Christian Offen . Backward error analysis for conjugate symplectic methods. Journal of Geometric Mechanics, 2023, 15(1): 98-115. doi: 10.3934/jgm.2023005 |
| [6] | Alexander Pigazzini, Cenap Özel, Saeid Jafari, Richard Pincak, Andrew DeBenedictis . A family of special case sequential warped-product manifolds. Journal of Geometric Mechanics, 2023, 15(1): 116-127. doi: 10.3934/jgm.2023006 |
| [7] | Álvaro Rodríguez Abella, Melvin Leok . Discrete Dirac reduction of implicit Lagrangian systems with abelian symmetry groups. Journal of Geometric Mechanics, 2023, 15(1): 319-356. doi: 10.3934/jgm.2023013 |
| [8] | Maulik Bhatt, Amit K. Sanyal, Srikant Sukumar . Asymptotically stable optimal multi-rate rigid body attitude estimation based on lagrange-d'alembert principle. Journal of Geometric Mechanics, 2023, 15(1): 73-97. doi: 10.3934/jgm.2023004 |
| [9] | Marcin Zając . The dressing field method in gauge theories - geometric approach. Journal of Geometric Mechanics, 2023, 15(1): 128-146. doi: 10.3934/jgm.2023007 |
| [10] | William Clark, Anthony Bloch . Existence of invariant volumes in nonholonomic systems subject to nonlinear constraints. Journal of Geometric Mechanics, 2023, 15(1): 256-286. doi: 10.3934/jgm.2023011 |
We derive the equations of motion of an action-dependent version of the Einstein-Hilbert Lagrangian as a specific instance of the Herglotz variational problem. Action-dependent Lagrangians lead to dissipative dynamics, which cannot be obtained with the standard method of Lagrangian field theory. First-order theories of this kind are relatively well understood, but examples of singular or higher-order action-dependent field theories are scarce. This work constitutes an example of such a theory. By casting the problem in clear geometric terms, we are able to obtain a Lorentz invariant set of equations, which contrasts with previous attempts.
| [1] |
K. Grabowska, J. Grabowski, A geometric approach to contact Hamiltonians and contact Hamilton–Jacobi theory, J. Phys. A-Math. Theor., 55 (2022), 435204. https://doi.org/10.1088/1751-8121/ac9adb doi: 10.1088/1751-8121/ac9adb
|
| [2] |
J. Gaset, X. Gràcia, M. Muñoz-Lecanda, X. Rivas, N. Román-Roy, New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries, Int. J. Geom. Methods. M., 17 (2020), 2050090. https://doi.org/10.1142/S0219887820500905 doi: 10.1142/S0219887820500905
|
| [3] |
A. Bravetti, Contact geometry and thermodynamics, Int. J. Geom. Methods. M., 16 (2019), 1940003. https://doi.org/10.1142/s0219887819400036 doi: 10.1142/s0219887819400036
|
| [4] |
R. Mrugala, J. D. Nulton, J. C. Schön, P. Salamon, Contact structure in thermodynamic theory, Rep. Math. Phys., 29 (1991), 109–121. https://doi.org/10.1016/0034-4877(91)90017-H doi: 10.1016/0034-4877(91)90017-H
|
| [5] |
A. A. Simoes, M. de León, M. L. Valcázar, D. M. de Diego, Contact geometry for simple thermodynamical systems with friction, P. Roy. Soc. A-Math. Phy., 476 (2020), 20200244. https://doi.org/10.1098/rspa.2020.0244 doi: 10.1098/rspa.2020.0244
|
| [6] |
F. M. Ciaglia, H. Cruz, G. Marmo, Contact manifolds and dissipation, classical and quantum, Ann. Phys-New. York., 398 (2018), 159–179. https://doi.org/10.1016/j.aop.2018.09.012 doi: 10.1016/j.aop.2018.09.012
|
| [7] |
S. I. Goto, Contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium statistical mechanics, J. Math. Phys., 57 (2016). https://doi.org/10.1063/1.4964751 doi: 10.1063/1.4964751
|
| [8] |
M. J. Lazo, J. Paiva, J. T. S. Amaral, G. S. F. Frederico, Action principle for action-dependent Lagrangians toward nonconservative gravity: Accelerating universe without dark energy, Phys. Rev. D., 95 (2017), 101501. https://doi.org/10.1103/PhysRevD.95.101501 doi: 10.1103/PhysRevD.95.101501
|
| [9] |
D. Sloan, New action for cosmology, Phys. Rev. D., 102 (2021), 043524. 10.1103/PhysRevD.103.043524 doi: 10.1103/PhysRevD.103.043524
|
| [10] |
J. Gaset, A. Marín-Salvador, Application of Herglotz's variational principle to electromagnetic systems with dissipation, Int. J. Geom. Methods. M., 19 (2022), 2250156. https://doi.org/10.1142/S0219887822501560 doi: 10.1142/S0219887822501560
|
| [11] | H. Geiges, An Introduction to Contact Topology, Cambridge University Press, 2008. https://doi.org/10.1017/CBO9780511611438 |
| [12] |
M. Lainz, M. de León, Contact Hamiltonian Systems, J. Math. Phys., 60 (2019), 102902. https://doi.org/10.1063/1.5096475 doi: 10.1063/1.5096475
|
| [13] | M. de León, M. Lainz, A review on contact Hamiltonian and Lagrangian systems, arXiv: 2011.05579 [math-ph]. |
| [14] |
M. de León, M. Lainz, Singular Lagrangians and precontact Hamiltonian systems, Int. J. Geom. Methods. M., 16 (2019), 1950158. https://doi.org/10.1142/S0219887819501585 doi: 10.1142/S0219887819501585
|
| [15] |
M. de León, J. Gaset, M. Laínz, M. C. Muñoz-Lecanda, N. Román-Roy, Higher-order contact mechanics, Ann. Phys-New. York., 425 (2021), 168396. https://doi.org/10.1016/j.aop.2021.168396 doi: 10.1016/j.aop.2021.168396
|
| [16] |
M. de León, J. Gaset, X. Gràcia, M. C. Muñoz-Lecanda, X. Rivas, Time-dependent contact mechanics, Monatsh. Math., (2022). https://doi.org/10.1007/s00605-022-01767-1 doi: 10.1007/s00605-022-01767-1
|
| [17] |
J. Gaset, X. Gràcia, M. C. Muñoz-Lecanda, X. Rivas, N. Román-Roy, A contact geometry framework for field theories with dissipation, Ann. Phys-New. York., 414 (2020), 168092. https://doi.org/10.1016/j.aop.2020.168092 doi: 10.1016/j.aop.2020.168092
|
| [18] |
J. Gaset, X. Gràcia, M. C. Muñoz-Lecanda, X. Rivas, N. Román-Roy, A K-contact Lagrangian formulation for nonconservative field theories, Rep. Math. Phys., 87 (2021), 347–368. https://doi.org/10.1016/S0034-4877(21)00041-02 doi: 10.1016/S0034-4877(21)00041-02
|
| [19] |
B. Georgieva, R. Guenther, T. Bodurov, Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem, J. Math. Phys., 44 (2003), 3911–3927. https://doi.org/10.1063/1.1597419 doi: 10.1063/1.1597419
|
| [20] | M. de León, J. Gaset, M. C. Muñoz-Lecanda, X. Rivas, N. Román-Roy, Multicontact formulation for non-conservative field theories, J. Phys. A-Math. Theor., (2023). |
| [21] |
G. J. Olmo, D. Rubiera-Garcia, A. Wojnar, Stellar structure models in modified theories of gravity: lessons and challenges, Phys. Rep., 876 (2020), 1–75. https://doi.org/10.1016/j.physrep.2020.07.001 doi: 10.1016/j.physrep.2020.07.001
|
| [22] |
T. Baker, E. Bellini, P. G. Ferreira, M. Lagos, J. Noller, I. Sawicki, Strong Constraints on Cosmological Gravity from GW170817 and GRB 170817A, Phys. Rev. Lett., 119 (2017), 1–6. https://doi.org/10.1103/PhysRevLett.119.251301 doi: 10.1103/PhysRevLett.119.251301
|
| [23] |
J. A. P. Paiva, M. J. Lazo, V. T. Zanchin, Generalized nonconservative gravitational field equations from Herglotz action principle, Phys. Rev. D., 105 (2022), 124023. https://doi.org/10.1103/PhysRevD.105.124023 doi: 10.1103/PhysRevD.105.124023
|
| [24] | A. Mas Dorca, A variational derivation of the field equations of an action-dependent Einstein-Hilbert Lagrangian, Universitat Autònoma de Barcelona. |
| [25] | G. Herglotz, Berührungstransformationen, Lectures at the University of Göttingen, 1930. |
| [26] | M. León de, M. Lainz, M. C. Muñoz-Lecanda, The Herglotz Principle and Vakonomic Dynamics, In: Geometric Science of Information, 12829 (2021), 183–190. https://doi.org/10.1007/978-3-030-80209-7-21 |
| [27] |
M. J. Lazo, J. Paiva, J. T. S. Amaral, G. S. F. Frederico, An action principle for action-dependent Lagrangians: Toward an action principle to non-conservative systems, J. Math. Phys., 59 (2018), 032902. https://doi.org/10.1063/1.5019936 doi: 10.1063/1.5019936
|
| [28] | G. Xavier, J. Marín-Solano, M. C. Muñoz-Lecanda, Some geometric aspects of variational calculus in constrained systems, Rep. Math. Phys., 51 (2003), 127–148. |
| [29] | J. Gaset, M. Lainz, A. Mas, X. Rivas, The Herglotz variational principle for dissipative field theories, arXiv: 2211.17058 [math-ph]. |
| [30] | S. M. Carroll, Lecture Notes on General Relativity, arXiv: gr-qc/9712019. |
| [31] |
J. Gaset, N. Román-Roy, Multisymplectic unified formalism for Einstein-Hilbert gravity, J. Math. Phys., 59 (2018). https://doi.org/10.1063/1.4998526 doi: 10.1063/1.4998526
|
| [32] |
M. E. Rosado, M. J. Muñoz, Second-order Lagrangians admitting a first-order Hamiltonian formalism, Ann. Mat. Pur. Appl., 197 (2018), 357–397. https://doi.org/10.1007/s10231-017-0683-y doi: 10.1007/s10231-017-0683-y
|
| [33] |
M. de León, J. Gaset, M. Lainz, Inverse problem and equivalent contact systems, J. Geom. Phys., 176 (2022), 104500. https://doi.org/10.1016/j.geomphys.2022.104500 doi: 10.1016/j.geomphys.2022.104500
|
| [34] |
G. Jordi R. Narciso, New symplectic approach to the Metric-Affine (Einstein-Palatini) action for gravity, J. Geom. Phys., 11 (2019), 361-396. https://doi.org/10.3934/jgm.2019019 doi: 10.3934/jgm.2019019
|
| [35] |
D. Vey, Multisymplectic formulation of vielbein gravity: Ⅰ. De Donder–Weyl formulation, Hamiltonian (n-1)-forms, Classical. Quant. Grav., 32 (2015), 095005. https://doi.org/10.1088/0264-9381/32/9/095005 doi: 10.1088/0264-9381/32/9/095005
|
| 1. | MOUHAMADOU SAMSIDY GOUDIABY, GUNILLA KREISS, A RIEMANN PROBLEM AT A JUNCTION OF OPEN CANALS, 2013, 10, 0219-8916, 431, 10.1142/S021989161350015X | |
| 2. | Jean-Marc Herard, 2012, The Coupling of Multi-phase Flow Models, 978-1-60086-933-4, 10.2514/6.2012-3357 | |
| 3. | Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch, Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure, 2021, 16, 1556-181X, 283, 10.3934/nhm.2021007 |
Jordi Gaset, Arnau Mas. A variational derivation of the field equations of an action-dependent Einstein-Hilbert Lagrangian[J]. Journal of Geometric Mechanics, 2023, 15(1): 357-374. doi: 10.3934/jgm.2023014
DownLoad: