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Jacobi equations and particle accelerator beam dynamics

Ricardo Gallego Torrome

Abstract

A geometric formulation of the linear beam dynamics in accelerator physics is presented. In particular, it is proved that the linear transverse and longitudinal dynamics can be interpret geometrically as an approximation to the Jacobi equation of an affine averaged Lorentz connection. We introduce a specific notion reference trajectory as integral curves of the main velocity vector field. A perturbation caused by the statistical nature of the bunch of particles is considered.

Jacobi equations and particle accelerator beam dynamics

Abstract

A geometric formulation of the linear beam dynamics in accelerator physics is presented. In particular, it is proved that the linear transverse and longitudinal dynamics can be interpret geometrically as an approximation to the Jacobi equation of an affine averaged Lorentz connection. We introduce a specific notion reference trajectory as integral curves of the main velocity vector field. A perturbation caused by the statistical nature of the bunch of particles is considered.

Paper Structure

This paper contains 16 sections, 4 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. The Lorentz connection and the averaged Lorentz connection
  3. Linear connections on pull-back bundles
  4. The Lorentz connection
  5. The averaged Lorentz connection
  6. Comparison between the geodesics of $\,^L\nabla$ and $\langle\,^L\nabla\rangle$
  7. The Jacobi equation of the averaged Lorentz dynamics
  8. Transversal beam dynamics from the Jacobi equation of the averaged Lorentz connection
  9. Transverse dynamics and the Jacobi equation of the averaged Lorentz connection
  10. Examples of transverse linear dynamics
  11. A perturbative collective dynamical effect
  12. Calculation of the dispersion function in beam dynamics
  13. An observable associated with the composed nature of the bunch of particles
  14. Longitudinal beam dynamics and corrections from the Jacobi equation of the averaged Lorentz connection
  15. Examples
  16. ...and 1 more sections

Key Result

Proposition 2.1

Given a semi-spray $G^i(x,y)$ there is defined a connection on $\pi^*{\bf TM}$ determined by the relations where $\{\pi^*e_i, \,i=0,1,2,3\,\}$ is a local frame for sections $\Gamma(\pi^*{\bf TM})$.

Theorems & Definitions (4)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 2.4