  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 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, 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 GEMPIC [Kraus et al.,
arXiv:1609.03053v1 [math.NA] (2016)], a particle in cell code that incorporates
Hamiltonian and geometrical structure preserving properties.
