Transformation of phase space densities under the coordinate changes of accelerator physics

In self-consistent modeling of many-particle systems it is convenient to solve the Maxwell equations for self-fields in coordinates based in the laboratory with time as the evolution variable, but conventional and also convenient to follow particle motion in Frenet-Serret coordinates referred to a reference orbit, with arclength along that orbit as the evolution variable. We refer to these two pictures as the laboratory system and beam system descriptions, while emphasizing that a Lorentz transformation is not involved; it is only a matter of two alternative descriptions of motion in one inertial frame. The problem then arises of how to express the laboratory system charge/current density for the Maxwell equations in terms of the phase space density described in the beam system. We find the exact expression, then make justified approximations to put the formula in a simple and practical form. Incidentally, we derive exact and approximate equations of motion in the different coordinates, without the use of the canonical formalism. The results have been applied in a study of coherent synchrotron radiation in bunch compressors.

Figure 1

Basic lab system setup.

Figure 2

Beam frame coordinates.

Figure 3

Relation of B1 and B2 densities. The B1 density on plane $u=u_1$ in extended phase space is related to the B2 density on $s=s_0$ by the evolution under $\phi$, represented as a curve connecting $(s_0,u_0,w_0)$ to $(s_1,u_1,w_1)$. Think of the $w$ dimension as perpendicular to the $(s,u)$ plane.