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

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