Hybrid symplectic integrators for relativistic particles in electric and magnetic fields

We develop a new implicit symplectic integrator and derive the first and second order integration algorithms for it. We combine this integrator with a Yoshida composition and with a Suzuki composition to get third order hybrid symplectic integrators. We test the algorithms on an exactly solvable example from McMillan (1950 Phys. Rev. 79 498) of a particle in an electromagnetic wave and find very good results, including long term stability and exact preservation of constants associated with symmetries. We discuss the circumstances in which the new algorithms may be useful.