This project explores the preservation of the Hamiltonian structure of Vlasov-Maxwell equations in a numerical method. The fundamental aspects of applying spectral type methods to infinite dimensional Hamiltonian systems so that the resulting semi-discrete (continuous in time) system exactly preserves the system Hamiltonian and some other conservation laws are being investigated. A functional approximation operator is defined to map the true infinite dimensional phase space to a finite dimensional numerical approximation, and an imbedding operator is defined to map this numerical phase space back to the exact phase space. Together these operators allow the Poisson bracket on the infinite dimensional phase space to be transferred to the numerical phase space; Hamiltonian dynamics can then be defined on the numerical phase space using a numerical approximation to the exact Hamiltonian. The resulting finite dimensional Hamiltonian system represents a semi-discrete numerical method that exactly conserves the system energy. Conditions necessary for the numerical conservation of Casimirs and momenta are also being derived. These techniques are also being extended to Hamiltonian systems which are not isolated and allow general boundary conditions. The application of these techniques to the Vlasov-Maxwell equations will allow numerical plasma kinetic theory models which exactly conserve energy, particles and momentum, even at a finite expansion in space, velocity and time.
