# Approximating Nonlinear Forces with Phase-Space Decoupling

## TL;DR

This paper introduces a phase-space decoupling method using virtual coordinates to efficiently approximate nonlinear forces in beam dynamics, bridging the gap between fast linear and slow multiparticle simulations.

## Contribution

The paper presents a novel phase-space decoupling technique with virtual coordinates to efficiently model nonlinear effects in beam tracking.

## Key findings

- Significant reduction in computational cost
- Effective approximation of nonlinear forces
- Bridges gap between linear and multiparticle simulations

## Abstract

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.