Approximating nonlinear forces with phase-space decoupling

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 ₀) where x ₀ 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.