A variational derivation of the field equations of an action-dependent Einstein-Hilbert Lagrangian

Jordi Gaset; Arnau Mas; Jordi Gaset; Arnau Mas

doi:10.3934/jgm.2023014

Journal of Geometric Mechanics

2023, Volume 15, Issue 1: 357-374. doi: 10.3934/jgm.2023014

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Research article

A variational derivation of the field equations of an action-dependent Einstein-Hilbert Lagrangian

Jordi Gaset ^{1
,
,},
Arnau Mas ²

1.
Department of Applied Mathematics, Technical University of Madrid, Madrid, Madrid, Spain
2.
Faculty of Physics, Ludwig-Maximilians-Universität Munich, Munich, Bavaria, Germany

Received: 06 October 2022 Revised: 31 January 2023 Accepted: 15 February 2023 Published: 03 March 2023
37K58, 70S05, 53D10, 35R01, 83D05

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.
- contact geometry,
- general relativity,
- variational principles,
- contact gravity,
- non-conservative system
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

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Abstract

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.

References

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