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2.1 Introduction

2.1.1 Coordinate system

Beam dynamics consists of the study of a bunch of particles, bunch among which one particle is labelled as the reference particle. The reference particle follows a predescribed path, which is the ideal path that the bunch would follow. Two frames are then defined. Cartesian coordinates are associated to the laboratory frame $\Sigma$ , with axis

and

, the later being the main direction of propagation for a linear accelerator. A second frame is defined, $\Sigma^*$ , which moves along the reference particle trajectory. The axis associated with $\Sigma^*$ , are the curvilinear axis

and

, following the reference particle in its motion, and forming a right handed coordinate system.

being the curvilinear coordinate, we define the curvature vector for the reference particle as:

$\displaystyle \v{k}=-\frac{\v{S}(s)}{ds}$

(2.1)

In this frame, transverse coordinates

of particles are expressed with respect to the reference particle coordinates. When using the equation of motion we have to be careful that the time of the particule is different than the time of the reference particule. As we are mainly studying linear accelerator,

and

will be most of the time the same, so that we might use the

coordinate instead of

in some local cases..

**Figure 2.1:** Curvilinear system of coordinates used in particle beam dynamics. The coordinates are given in a frame $\Sigma^{*}$ that moves along the reference trajectory.

We will try to remember in this section that coordinates used further are coordinates relative to the reference particule. As descibed in figure 2.2, when we will write $x'_{in}$ , in fact it will correspond to a $dx'_{in}$ and it is the same for the output.

**Figure 2.2:** Illustration of the relative coordinates. For beam dynamics and phase space, coordinates are always relative to the reference particle.

2.1.2 Equation of motion

We will not establish in this paper the equations of motion, but we strongly recommand the reading of the section 4.2 and 4.7 from H. Wiedemann's book [43]. Nevertheless, I will present the resolution method I used, because it really helped me for my understanding of linear formalism, and according to me, justifies immediatly the use of matrix formalism. From the previously cited reference, the perturbated linear equation of motion is:

$\displaystyle f(s) = u''+ K(s).u$

(2.2)

where u stands for x or y, and s is the longitudinal coordinate. We chose to introduce since the beginning the dispertive term

, which is the illustration that particles in a beam have different deflections due to their different energies. The finite energy spread, or chromatic error can be at a first order assessed with the term $f(s)=\delta /\rho$ , where $\delta=\Delta p / p$ . The resolution of equation 2.2 will follow in the next sections.

Emmanuel Branlard
http://emmanuel.branlard.free.fr/