  We consider a general discretization strategy for Hamiltonian field theories
generated by Lie-Poisson brackets which we call dual PIC (DPIC). This method
involves prescribing two different discrete representations of the dynamical
variable which are constrained as a Casimir invariant of the flow to coincide
with one another via an L2 projection throughout the entire simulation. This
allows one to leverage the relative advantages of each discrete representation.
We begin by describing DPIC as applied to a general Lie-Poisson system and then
provide illustrative examples: the discretization of the two-dimensional
vorticity equations and the Vlasov-Poisson equation.
