A Tutorial on Variational Integrators
  We present a brief tutorial on the nuts and bolts computation of a multisymplectic particle-in-cell algorithm using the discretized Lagrangian approach. This approach, originated by Marsden, Shadwick, and others, brings the benefits of symplectic integration of Hamiltonian systems to full electromagnetic particle-in-cell algorithms. To make the work more approachable, we present a basic discussion of the philosophy, combined with a detailed derivation of a standard 1-dimensional electrostatic particle-in-cell algorithm. 
Stephen D. Webb
physics.plasm-ph