Energy and momentum conservation in the Euler–Poincaré formulation of local Vlasov–Maxwell-type systems. (19th May 2020)

• Record Type: Journal Article
• Authors: Hirvijoki, Eero; Burby, Joshua W; Pfefferlé, David; Brizard, Alain J
• Is Part Of: Journal of physics. Volume 53:Number 23(2020)
• Publication Date: 2020-05-19
• Subjects: Euler–Poincaré reduction -- Noether theorem -- plasma physics -- kinetic theory; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/ab8b38
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: 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.