Symplectic Integration of Magnetic Systems

Dependable numerical results from long-time simulations require stable numerical integration schemes. For Hamiltonian systems, this is achieved with symplectic integrators, which conserve the symplectic condition and exactly solve for the dynamics for an approximate Hamiltonian. The conventional generating function formalism used for Hamiltonian systems is problematic for magnetic systems, as the Hamiltonian is not easily separable. This paper presents a derivation of symplectic integrators from a discretized action. Using this method, it is shown that the integrator due to Boris is a symplectic integrator in the non-relativistic limit, and a relativistic integrator is derived.