Variational Formulation of E & M Particle Simulation Algorithms in   Cylindrical Geometry using an Angular Modal Decomposition
  Taking advantage of the flexibility of the variational method with coordinate transformations, we derive a self-consistent set of equations of motion from a discretized Lagrangian to study kinetic plasmas using a Fourier decomposed cylindrical coordinate system. The phase-space distribution function was reduced to a collection of finite-sized macro-particles of arbitrary shape moving on a virtual Cartesian grid. However, the discretization of field quantities was performed in cylindrical coordinates and decomposed into a truncated Fourier series in angle. A straightforward finite element interpolation scheme is used to transform between the two grids. The equations of motion were then obtained by demanding the action be stationary. The primary advantage of the variational approach is preservation of Lagrangian symmetries. In the present case, this leads to exact energy conservation, thus avoiding possible difficulties with grid heating. 
A. B. Stamm and B. A. Shadwick
physics.comp-ph physics.plasm-ph