The Hamiltonian structure and Euler-Poincar\'{e} formulation of the Vlasov-Maxwell and gyrokinetic systems

TL;DR

This paper develops a new variational and Hamiltonian framework for the gyrokinetic system, enabling advanced geometric and stability analyses in plasma physics.

Contribution

It introduces a novel Eulerian variational principle and Hamiltonian structure for gyrokinetics, extending the theoretical foundation of plasma models.

Findings

Derived a new variational principle for gyrokinetics.

Explicitly formulated the Hamiltonian structure using a modified Dirac constraint theory.

Potential applications include geometric integration, large-eddy simulation, and stability analysis.

Abstract

We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in Ref. 1. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with a modified Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincar\'{e} theory. Possible applications of these formulations include continuum geometric integration techniques, large-eddy simulation models and Casimir type stability methods. [1] H. Cendra et. al., Journal of Mathematical Physics 39, 3138 (1998)