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

Stephan I. Tzenov

TL;DR

This work evaluates the Hamilton-Jacobi equation as a practical tool for nonlinear beam dynamics by constructing a full canonical transformation for a thin sextupole and showing its equivalence to the Hénon map. It reveals a subtle but important discrepancy between solving with the nonlinear kick preceding the one-turn rotation and the reverse sequence, yielding a standard versus a backward Henon map, respectively. A generalized twist map is developed for combined sextupole and octupole nonlinearities, and a Liouville-based statistical description is used to study beam density evolution and resonance-induced holes. Together, these results clarify how Hamilton-Jacobi methods relate to standard symplectic maps and illuminate long-term stability and density redistribution in nonlinear accelerator lattices.

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.

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

TL;DR

This work evaluates the Hamilton-Jacobi equation as a practical tool for nonlinear beam dynamics by constructing a full canonical transformation for a thin sextupole and showing its equivalence to the Hénon map. It reveals a subtle but important discrepancy between solving with the nonlinear kick preceding the one-turn rotation and the reverse sequence, yielding a standard versus a backward Henon map, respectively. A generalized twist map is developed for combined sextupole and octupole nonlinearities, and a Liouville-based statistical description is used to study beam density evolution and resonance-induced holes. Together, these results clarify how Hamilton-Jacobi methods relate to standard symplectic maps and illuminate long-term stability and density redistribution in nonlinear accelerator lattices.

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.
Paper Structure (9 sections, 59 equations, 5 figures)

This paper contains 9 sections, 59 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The Classical Hamilton-Jacobi Equation
  3. Perturbation Solution of the Hamilton-Jacobi Equation
  4. Hamilton's Equations of Motion in Interaction Representation and the Hénon Map
  5. Hamilton's Equations of Motion
  6. The Standard and the Backward Henon Map
  7. The General Twist Map
  8. Statistical Description of Nonlinear Dynamics
  9. Concluding Remarks and Outlook

Figures (5)

  • Figure 1: Phase portrait of the canonical Hénon map \ref{['CanonHenMap']} close to the third-order nonlinear resonance $3 \nu = {\rm integer}$. The particular value of the fractional part of the unperturbed betatron tune is taken to be $0.31114$.
  • Figure 2: Phase portrait of the backward canonical Hénon map counterpart of Eqs. \ref{['RevHenMap']} close to the third-order resonance $3 \nu = {\rm integer}$. Similar to Fig. \ref{['fig:1']}, the particular value of the fractional part of the unperturbed betatron tune is taken to be $0.31114$.
  • Figure 3: Phase portrait of the generalized twist map of Eqs. \ref{['GenTwistMap']} close to the fourth-order resonance $4 \nu = {\rm integer}$. The octupole location is taken to be $\theta_o = \pi$, while its relative strength is $\Lambda_o \sim 1$. The particular value of the fractional part of the unperturbed betatron tune is taken to be $0.243$.
  • Figure 4: Unperturbed density profile $\varrho_0 {\left( X \right)}$ according to Eq. \ref{['LinearSol']}.
  • Figure 5: Density profile $\varrho_0 {\left( X, \chi \right)}$, where ${\mathcal{J}}$ has been replaced by ${\mathcal{J}} + {\mathcal{J}}_1$ according to Eq. \ref{['TentSolF']}. The particular value of the fractional part of the unperturbed betatron tune is taken to be $0.3333265$, that is close to the third-order resonance.