Canonical Variables Describing Orbits

As from Version 9.01, MAD uses the following canonical variables to describe the motion of particles:

X

Horizontal position $x$ of the (closed) orbit, referred to the ideal orbit [m].

PX

Horizontal canonical momentum of the (closed) orbit referred to the ideal orbit, divided by the reference momentum: $\mathtt{PX} = p_x / p_0$.

Y

Vertical position $y$ of the (closed) orbit, referred to the ideal orbit [m].

PY

Vertical canonical momentum of the (closed) orbit referred to the ideal orbit, divided by the reference momentum: $\mathtt{PY} = p_y / p_0$.

T

The negative time difference, multiplied by the instantaneous velocity of the particle [m]: $\mathtt{T} = - v \delta(t)$. A positive T means that the particle arrives ahead of the reference particle. T describes the deviation of the particle from the orbit of a fictitious reference particle having the constant reference momentum $p_s$ and the reference velocity $v_s$. $v_s$ defines the revolution frequency. The velocities have the values

$$v = c p / \sqrt{p^2 + m^2 c^2}, \qquad v_s = c p_s / \sqrt{p_s^2 + m^2 c^2},$$

where $c$ is the velocity of light, $m$ is the particle rest mass, and $p$ is the instantaneous momentum of the particle.

PT

Momentum error, divided by the reference momentum: $\mathtt{PT} = \delta p / p_s$. This value is only non-zero when synchrotron motion is present. It describes the deviation of the particle from the orbit of a particle with the reference momentum $p_s$.

The independent variable is:

S

Arc length $s$ along the reference orbit, [m].

The longitudinal variables have been changed with respect to previous versions of MAD, so as to be in line with the CLASSIC project. In the limit of fully relativistic particles ( $\gamma \gg 1, v = c, p c = E$), the variables T, PT used here agree with the longitudinal variables used in TRANSPORT. This means that T becomes the negative path length difference, while PT is the fractional momentum error. The reference momentum must be constant in order to keep the system canonical.