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An action principle for the Vlasov equation and associated Lie perturbation equations. Part 1. The Vlasov—Poisson system

Published online by Cambridge University Press:  13 March 2009

Jonas Larsson
Jonas Larsson
Affiliation:
Department of Plasma Physics, Umeå University, S-90187 Umeå, Sweden, and Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720, U.S.A.
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Abstract

A new action principle determining the dynamics of the Vlasov–Poisson system is presented (the Vlasov–Maxwell system will be considered in Part 2). The particle distribution function is explicitly a field to be varied in the action principle, in which only fundamentally Eulerian variables and fields appear. The Euler–Lagrange equations contain not only the Vlasov–Poisson system but also equations associated with a Lie perturbation calculation on the Vlasov equation. These equations greatly simplify the extensive algebra in the small-amplitude expansion. As an example, a general, manifestly Manley–Rowesymmetric, expression for resonant three-wave interaction is derived. The new action principle seems ideally suited for the derivation of action principles for reduced dynamics by the use of various averaging transformations (such as guiding-centre, oscillation-centre or gyro-centre transformations). It is also a powerful starting point for the application of field-theoretical methods. For example, the recently found Hermitian structure of the linearized equations is given a very simple and instructive derivation, and so is the well-known Hamiltonian bracket structure of the Vlasov–Poisson system.

Type
Research Article
Information
Journal of Plasma Physics , Volume 48 , Issue 1 , August 1992 , pp. 13 - 35
Copyright
Copyright © Cambridge University Press 1992

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