Explicit symplectic algorithms based on generating functions for   relativistic charged particle dynamics in time-dependent electromagnetic   field
  Relativistic dynamics of a charged particle in time-dependent electromagnetic fields has theoretical significance and a wide range of applications. It is often multi-scale and requires accurate long-term numerical simulations using symplectic integrators. For modern large-scale particle simulations in complex, time-dependent electromagnetic field, explicit symplectic algorithms are much more preferable. In this paper, we treat the relativistic dynamics of a particle as a Hamiltonian system on the cotangent space of the space-time, and construct for the first time explicit symplectic algorithms for relativistic charged particles of order 2 and 3 using the sum-split technique and generating functions. 
Ruili Zhang, Yulei Wang, Yang He, Jianyuan Xiao, Jian Liu, Hong Qin   and Yifa Tang
physics.plasm-ph