Hamiltonian and action principle (HAP) formulations of plasma physics are reviewed for the purpose of explaining structure preserving numerical algorithms. Geometric structures associated with and emergent from HAP formulations are discussed. These include conservative integration, which exactly conserves invariants, symplectic integration, which exactly preserves the Hamiltonian geometric structure, and other Hamiltonian integration techniques. Basic ideas of variational integration and Poisson integration, which can preserve the noncanonical Hamiltonian structure, are discussed. Metriplectic integration, which preserves the structure of conservative systems with both Hamiltonian and dissipative parts, is proposed. Two kinds of simulated annealing, a relaxation technique for obtaining equilibrium states, are reviewed: one that uses metriplectic dynamics, which maximizes an entropy at fixed energy, and the other that uses double bracket dynamics, which preserves Casimir invariants. Throughout, applications to plasma systems are emphasized. The paper culminates with a discussion of geometric electromagnetic particle-in-cell [Kraus et al., J. Plasma Phys. (to be published); e-print arXiv:1609.03053v1 [math.NA]], a particle in cell code that incorporates Hamiltonian and geometrical structure preserving properties.

Bracket formulations of two kinds of dynamical systems, called incomplete and complete, are reviewed and developed, including double bracket and metriplectic dynamics. Dissipation based on the Cartan-Killing metric is introduced. Various examples of incomplete and complete dynamics are discussed, including dynamics associated with three-dimensional Lie algebras.

There are several plasma models intermediate in complexity between ideal magnetohydrodynamics (MHD) and two-fluid theory, with Hall and Extended MHD being two important examples. In this paper, we investigate several aspects of these theories, with the ultimate goal of deriving the noncanonical Poisson brackets used in their Hamiltonian formulations. We present fully Lagrangian actions for each, as opposed to the fully Eulerian, or mixed Eulerian-Lagrangian, actions that have appeared previously. As an important step in this process, we exhibit each theory's two advected fluxes (in analogy to ideal MHD's advected magnetic flux), discovering also that with the correct choice of gauge they have corresponding Lie-dragged potentials resembling the electromagnetic vector potential, and associated conserved helicities. Finally, using the Euler-Lagrange map, we show how to derive the noncanonical Eulerian brackets from canonical Lagrangian ones.

An explicit high-order noncanonical symplectic algorithm for ideal two-fluid systems is developed. The fluid is discretized as particles in the Lagrangian description, while the electromagnetic fields and internal energy are treated as discrete differential form fields on a fixed mesh. With the assistance of Whitney interpolating forms [H. Whitney, Geometric Integration Theory (Princeton University Press, 1957); M. Desbrun et al., Discrete Differential Geometry (Springer, 2008); J. Xiao et al., Phys. Plasmas 22, 112504 (2015)], this scheme preserves the gauge symmetry of the electromagnetic field, and the pressure field is naturally derived from the discrete internal energy. The whole system is solved using the Hamiltonian splitting method discovered by He et al. [Phys. Plasmas 22, 124503 (2015)], which was been successfully adopted in constructing symplectic particle-in-cell schemes [J. Xiao et al., Phys. Plasmas 22, 112504 (2015)]. Because of its structure preserving and explicit nature, this algorithm is especially suitable for large-scale simulations for physics problems that are multi-scale and require long-term fidelity and accuracy. The algorithm is verified via two tests: studies of the dispersion relation of waves in a two-fluid plasma system and the oscillating two-stream instability.

The noncanonical Hamiltonian formulation of magnetohydrodynamics (MHD) is used to construct variational principles for continuously symmetric equilibrium configurations of magnetized plasma, including flow. In particular, helical symmetry is considered, and results on axial and translational symmetries are retrieved as special cases of the helical configurations. The symmetry condition, which allows the description in terms of a magnetic flux function, is exploited to deduce a symmetric form of the noncanonical Poisson bracket of MHD. Casimir invariants are then obtained directly from the Poisson bracket. Equilibria are obtained from an energy-Casimir principle and reduced forms of this variational principle are obtained by the elimination of algebraic constraints.