#### Symplectic methods for numerical integration

Symplectic or geometric integrators are numerical schemes designed around the mathematical properties of the underlying physical system, often by making the integrator a discrete Hamiltonian system. These integrators often have very nice conservation properties, are stable, and generally nice to work with. I’ve worked on a variety of these integrators, most notably a particle-in-cell code that is derived from a single space-time Lagrangian and conserves current by discretely satisfying gauge invariance. (Collaborators: Hong Qin, Josh Burby, Bill Tang, Lee Ellison)

#### Hamiltonian structure of plasma physics

The Hamiltonian structure of physical theories can be fascinating in its own right, and plasma physics presents many examples of theories with different mathematical structures. For instance, it is not always possible to find canonical variables (variables in which Hamilton’s equations take their usual form) even if a system is Hamiltonian. I’ve mainly focused on gyrokinetics and guiding center theory, which are theories in which the fast gyromotion of particles in “averaged out” to obtain a system of equations that is easier to understand and solve. (Collaborators: Hong Qin, Josh Burby).

#### Guiding center equations

The guiding center equations have been a mainstay of plasma physics for 50 years, since they allow one to consider the dynamics of particles without having to resolve the extremely fast gyromotion. Hamiltonian dynamics can be very useful for deriving these at higher order, however things become very complex. This motivated Josh Burby and I to write the first system for deriving these automatically in on a computer. This also resulted in the VEST package, which quickly and easily performs abstract vector calculus simplifications. (Collaborators: Josh Burby, Hong Qin)