We consider Hamiltonian particle mesh methods [1], in which Lagrangian fluid or plasma parcels are discretized as particles, whose dynamics are determined via Eulerian potentials considered on a mesh or on finite elements. Hamiltonian particles mesh methods for fluid or plasma systems have hybrid Eulerian-Lagrangian formulations as continuum counterparts [2].

For a sample system, Vlasov-Poisson plasma dynamics, we derive a constrained Eulerian-Lagrangian variational principle. The constraint is used to couple the Lagrangian and Eulerian parts of the dynamics. This variational principle can subsequently serve as a starting point for several discretizations. These include particles with splines or Bernstein polynomials as compact support, with the Eulerian potential calculated on finite difference or finite element meshes. Furthermore, we constructed hierarchical basis functions to accommodate multi-scale modelling. Numerical results suggest that modified basis functions on a fixed, coarse mesh might close the Eulerian fine-scale interactions. Due to the geometric (Hamiltonian) nature of the discrete dynamics, conservation is reached by construction. Given this closure of the Eulerian part of the dynamics, we discuss approaches for dealing with thermalization, by approximating the unresolved Lagrangian particle dynamics.