The Hamilton-Jacobi Equation and its Application to Nonlinear Beam Dynamics: Comparison of Approaches

TL;DR

This paper explores the Hamilton-Jacobi equation's application to nonlinear beam dynamics, comparing different approaches to modeling particle trajectories and analyzing the statistical behavior of particle beams under sextupole influence.

Contribution

It provides a comparative analysis of Hamilton-Jacobi-based methods for nonlinear beam dynamics and highlights an overlooked peculiarity in the order of kicks and rotations.

Findings

Difference between forward and backward twist maps depending on the order of operations.

The nontrivial peculiarity affects the resulting beam dynamics models.

Statistical properties of particle beams under sextupole influence are characterized.

Abstract

The rarely used Hamilton-Jacobi equation has been utilized as an elegant way to find the trajectories of mechanical systems and to derive symplectic maps. Further, the exact solution in kick approximation of Hamilton's equations of motion in interaction representation is written as a generalized one-turn twist map. One can imagine that the nonlinear kick comes first, followed by the one-period rotation along the machine circumference, or a second alternative in which the one-period rotation occurs before the kick. There is a difference in the result of solving Hamilton's equations between the two cases, which is expressed in obtaining a standard forward twist map in the first case, or alternatively a backward map in the second one. This nontrivial and intuitively unclear peculiarity is usually ignored/overlooked in practically all specialized references on the topic. Finally, the statistical properties and the behavior of the density distribution of a particle beam in configuration space under the influence of an isolated sextupole have been studied.