  Beam tracking software for accelerators typically falls into two categories:
fast envelope simulations limited to linear beam optics, and slower
multiparticle simulations that can model nonlinear effects. To find a middle
ground between these approaches, we introduce virtual coordinates in position
and momentum which have a cross-dependency (i.e. p=f(x) where x is an initial
position and p* is a virtual projection of momentum onto the position axis).
This technique approximates multiparticle simulations with a significant
reduction in calculation cost.
