Abstract
The single-particle dynamics in a time-dependent focusing field is examined. The existence of the Courant-Snyder invariant, a fundamental concept in accelerator physics, is fundamentally a result of the corresponding symmetry admitted by the harmonic oscillator equation with linear time-dependent frequency. It is demonstrated that the Lie algebra of the symmetry group for the oscillator equation with time-dependent frequency is eight dimensional, and is composed of four independent subalgebras. A detailed analysis of the admitted symmetries reveals a deeper connection between the nonlinear envelope equation and the oscillator equation. A general theorem regarding the symmetries and invariants of the envelope equation, which includes the existence of the Courant-Snyder invariant as a special case, is demonstrated. As an application to accelerator physics, the symmetries of the envelope equation enable a fast numerical algorithm for finding matched solutions without using the conventional iterative Newton's method, where the envelope equation needs to be numerically integrated once for every iteration, and the Jacobi matrix needs to be calculated for the envelope perturbation.
Highlights
The Courant-Snyder invariant for an oscillator equation with time-dependent frequency is an important concept in accelerator physics [1,2], and is a powerful constraint used to characterize the motion of a charged particle in alternating-gradient field configurations
We investigate the symmetry group of the envelope equation itself
It was shown that the symmetry group for the oscillator equation with time-dependent frequency is generated by an 8D Lie algebra, which contains a subalgebra gb corresponding to the Courant-Snyder invariants
Summary
The Courant-Snyder invariant for an oscillator equation with time-dependent frequency is an important concept in accelerator physics [1,2], and is a powerful constraint used to characterize the motion of a charged particle in alternating-gradient field configurations. I const (independent of t), and w wt is any solution of the envelope equation w tw This classical result has been derived many times using different methods. It was derived by Courant and Snyder in 1958 [1] using the basic techniques for solving Hill’s equation It was rediscovered by Lewis [4] using the asymptotic method developed by Kruskal [5]; Eliezer and Gray [6] demonstrated a physical interpretation of the invariant; a derivation using a linear canonical transformation was given by Leach [7]; and Lutzky rederived the result using Noether’s theorem [8]. Newton’s method, where the envelope equation needs to be numerically integrated once for every iteration, and the
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