  Studying single-particle dynamics over many periods of oscillations is a
well-understood problem solved using symplectic integration. Such integration
schemes derive their update sequence from an approximate Hamiltonian,
guaranteeing that the geometric structure of the underlying problem is
preserved. Simulating a self-consistent system over many oscillations can
introduce numerical artifacts such as grid heating. This unphysical heating
stems from using non-symplectic methods on Hamiltonian systems. With this
guidance, we derive an electrostatic algorithm using a discrete form of
Hamilton's Principle. The resulting algorithm, a gridless spectral
electrostatic macroparticle model, does not exhibit the unphysical heating
typical of most particle-in-cell methods. We present results of this using a
two-body problem as an example of the algorithm's energy- and
momentum-conserving properties.
