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Hamiltonian structure is pursued and uncovered in collisional andcollisionless gyrokinetic theory. A new Hamiltonian formulation ofcollisionless electromagnetic theory is presented that is ideally suited toimplementation on modern supercomputers. The method used to uncover thisstructure is described in detail and applied to a number of examples, whereseveral well-known plasma models are endowed with a Hamiltonian structure forthe first time. The first energy- and momentum-conserving formulation of full-Fcollisional gyrokinetics is presented. In an effort to understand thetheoretical underpinnings of this result at a deeper level, a \emph{stochastic}Hamiltonian modeling approach is presented and applied to pitch anglescattering. Interestingly, the collision operator produced by the Hamiltonianapproach is equal to the Lorentz operator plus higher-order terms, but does notexactly conserve energy. Conversely, the classical Lorentz collision operatoris provably not Hamiltonian in the stochastic sense.

Chasing Hamiltonian structure in gyrokinetic theory