Energy and momentum conservation in the Euler–Poincaré formulation of local Vlasov–Maxwell-type systems
Abstract
The action principle by Low for the classic Vlasov–Maxwell system contains a mix of Eulerian and Lagrangian variables. This renders the Noether analysis of reparametrization symmetries inconvenient, especially since the well-known energy- and momentum-conservation laws for the system are expressed in terms of Eulerian variables only. While an Euler–Poincaré formulation of Vlasov–Maxwell-type systems, effectively starting with Low's action and using constrained variations for the Eulerian description of particle motion, has been known for a while Cendra et al, it is hard to come by a documented derivation of the related energy- and momentum-conservation laws in the spirit of the Euler–Poincaré machinery. To our knowledge only one such derivation exists in the literature so far, dealing with the so-called guiding-center Vlasov–Darwin system Sugama et al. The present exposition discusses a generic class of local Vlasov–Maxwell-type systems, with a conscious choice of adopting the language of differential geometry to exploit the Euler–Poincaré framework to its full extent. After reviewing the transition from a Lagrangian picture to an Eulerian one, we demonstrate how symmetries generated by isometries in space lead to conservation laws for linear- and angular-momentum density and how symmetry by time translation produces a conservation law for energy density. Here, wemore »
- Authors:
-
[1];
[2];
[3];
- Aalto Univ., Otaniemi (Finland)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Univ. of Western Australia, Perth, WA (Australia)
- Saint Michael's College, Colchester, VT (United States)
- Publication Date:
- Research Org.:
- Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
- Sponsoring Org.:
- USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA); Academy of Finland; National Science Foundation (NSF)
- OSTI Identifier:
- 1734726
- Report Number(s):
- LA-UR-19-32412
Journal ID: ISSN 1751-8113; TRN: US2205181
- Grant/Contract Number:
- 89233218CNA000001; 20180756PRD4; 315278; PHY-1805164
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Physics. A, Mathematical and Theoretical
- Additional Journal Information:
- Journal Volume: 53; Journal Issue: 23; Journal ID: ISSN 1751-8113
- Publisher:
- IOP Publishing
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Mathematics; Magnetic Fusion Energy
Citation Formats
Hirvijoki, Eero, Burby, Joshua W., Pfefferlé, David, and Brizard, Alain J. Energy and momentum conservation in the Euler–Poincaré formulation of local Vlasov–Maxwell-type systems. United States: N. p., 2020.
Web. doi:10.1088/1751-8121/ab8b38.
Hirvijoki, Eero, Burby, Joshua W., Pfefferlé, David, & Brizard, Alain J. Energy and momentum conservation in the Euler–Poincaré formulation of local Vlasov–Maxwell-type systems. United States. https://doi.org/10.1088/1751-8121/ab8b38
Hirvijoki, Eero, Burby, Joshua W., Pfefferlé, David, and Brizard, Alain J. Tue .
"Energy and momentum conservation in the Euler–Poincaré formulation of local Vlasov–Maxwell-type systems". United States. https://doi.org/10.1088/1751-8121/ab8b38. https://www.osti.gov/servlets/purl/1734726.
@article{osti_1734726,
title = {Energy and momentum conservation in the Euler–Poincaré formulation of local Vlasov–Maxwell-type systems},
author = {Hirvijoki, Eero and Burby, Joshua W. and Pfefferlé, David and Brizard, Alain J.},
abstractNote = {The action principle by Low for the classic Vlasov–Maxwell system contains a mix of Eulerian and Lagrangian variables. This renders the Noether analysis of reparametrization symmetries inconvenient, especially since the well-known energy- and momentum-conservation laws for the system are expressed in terms of Eulerian variables only. While an Euler–Poincaré formulation of Vlasov–Maxwell-type systems, effectively starting with Low's action and using constrained variations for the Eulerian description of particle motion, has been known for a while Cendra et al, it is hard to come by a documented derivation of the related energy- and momentum-conservation laws in the spirit of the Euler–Poincaré machinery. To our knowledge only one such derivation exists in the literature so far, dealing with the so-called guiding-center Vlasov–Darwin system Sugama et al. The present exposition discusses a generic class of local Vlasov–Maxwell-type systems, with a conscious choice of adopting the language of differential geometry to exploit the Euler–Poincaré framework to its full extent. After reviewing the transition from a Lagrangian picture to an Eulerian one, we demonstrate how symmetries generated by isometries in space lead to conservation laws for linear- and angular-momentum density and how symmetry by time translation produces a conservation law for energy density. Here, we also discuss what happens if no symmetries exist. Finally, two explicit examples will be given—the classic Vlasov–Maxwell and the drift-kinetic Vlasov–Maxwell—and the results expressed in the language of regular vector calculus for familiarity.},
doi = {10.1088/1751-8121/ab8b38},
journal = {Journal of Physics. A, Mathematical and Theoretical},
number = 23,
volume = 53,
place = {United States},
year = {Tue May 19 00:00:00 EDT 2020},
month = {Tue May 19 00:00:00 EDT 2020}
}
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