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

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.\\[4pt] [1] H. Cendra, D. D. Holm, M. J. W. Hoyle, and J. E. Marsden, Journal of Mathematical Physics 39, 3138 (1998).