2020
1.
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 Journal Article
In: Journal of Physics A: Mathematical and Theoretical, vol. 53, no. 23, pp. 235204, 2020.
Abstract | Links | BibTeX | Tags: drift-kinetic, euler-poincaré, Hamiltonian, vlasov-maxwell
@article{hirvijoki-2020,
title = {Energy and momentum conservation in the Euler\textendashPoincar\'{e} formulation of local Vlasov\textendashMaxwell-type systems},
author = {Eero Hirvijoki and Joshua W Burby and David Pfefferl\'{e} and Alain J Brizard},
url = {https://doi.org/10.1088%2F1751-8121%2Fab8b38},
doi = {10.1088/1751-8121/ab8b38},
year = {2020},
date = {2020-05-01},
journal = {Journal of Physics A: Mathematical and Theoretical},
volume = {53},
number = {23},
pages = {235204},
publisher = {IOP Publishing},
abstract = {The action principle by Low (1958 Proc. R. Soc. Lond. A 248 282\textendash7) for the classic Vlasov\textendashMaxwell 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\textendashPoincar\'{e} formulation of Vlasov\textendashMaxwell-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 (1998 J. Math. Phys. 39 3138\textendash57), it is hard to come by a documented derivation of the related energy- and momentum-conservation laws in the spirit of the Euler\textendashPoincar\'{e} machinery. To our knowledge only one such derivation exists in the literature so far, dealing with the so-called guiding-center Vlasov\textendashDarwin system Sugama et al (2018 Phys. Plasmas 25 102506). The present exposition discusses a generic class of local Vlasov\textendashMaxwell-type systems, with a conscious choice of adopting the language of differential geometry to exploit the Euler\textendashPoincar\'{e} 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. We also discuss what happens if no symmetries exist. Finally, two explicit examples will be given\textemdashthe classic Vlasov\textendashMaxwell and the drift-kinetic Vlasov\textendashMaxwell\textemdashand the results expressed in the language of regular vector calculus for familiarity.},
keywords = {drift-kinetic, euler-poincar\'{e}, Hamiltonian, vlasov-maxwell},
pubstate = {published},
tppubtype = {article}
}
The action principle by Low (1958 Proc. R. Soc. Lond. A 248 282–7) 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 (1998 J. Math. Phys. 39 3138–57), 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 (2018 Phys. Plasmas 25 102506). 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. 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.