Energy and momentum conservation in the Euler–Poincaré formulation of local Vlasov–Maxwell-type systems
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.
- Research Organization:
- Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
- Sponsoring Organization:
- USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA); Academy of Finland; National Science Foundation (NSF)
- Grant/Contract Number:
- 89233218CNA000001; 20180756PRD4; 315278; PHY-1805164
- OSTI ID:
- 1734726
- Report Number(s):
- LA-UR-19-32412; TRN: US2205181
- Journal Information:
- Journal of Physics. A, Mathematical and Theoretical, Vol. 53, Issue 23; ISSN 1751-8113
- Publisher:
- IOP PublishingCopyright Statement
- Country of Publication:
- United States
- Language:
- English
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