Description of particles using co-moving frame in accelerator I am learning particle accelerators and there is a basic concept as to description of particle beams in accelerators which confused me a lot.
Moving orthogonal, right-handed coordinate system (x, y, s) that follows a reference particle traveling along its ideal path (design orbit) is defined like this:

So, in order to describe any particle in the accelerator we must relate its position $\vec{r}$ to an ideal particle which has ideal energy and orbit $\vec{r_0} $.
However, in the literature i found in any books or documents of CERN, it always assumes the particle moves co-planner with the ideal particle, which is impossible i think.
In the book The Physics of Particle Accelerators written by Klaus Wille. He adopted such description, and he used it to derive anything.





As we can see in Eq(3.9) he neglect the $\vec{s_0}$ component of a general particle.
Why is that?
There are many resources in CAS which also indicate the case(page 30).
https://cas.web.cern.ch/sites/cas.web.cern.ch/files/lectures/constanta-2018/tlbd.pdf
 A: The short answer is that the terms that arise from keeping the $\hat{s_{0}}$ term are typically small for traditional accelerators.
The longer answer is just keeping the terms and seeing what shakes out.  We start with the quantity we want: $\vec{r} - \vec{r_{0}}$ which is the motion of a particle with respect to the design particle (equation 3-9):
\begin{equation}
\vec{r} - \vec{r}_0 = x\hat{x_0} + z\hat{z_0} + q\hat{s_0}.
\end{equation}
Here q is the coordinate the defines the separation of the particle of interest from the reference particle.
Now take the time derivative and make all the substitutions that Wille defines and you get,
\begin{equation}
\dot{\vec{r}} - \dot{\vec{r}_0} = \dot{x} \hat{x_{0}} + x \frac{\dot{s}}{R}\hat{s_0} + \dot{z}\hat{z_0} + \dot{q}\hat{s_{0}} - q\frac{\dot{s}}{R}\hat{x_0}.
\end{equation}
Collecting all the terms by unit vector and making a few more substitutions that Wille does you find,
\begin{equation}
\dot{\vec{r}} = \left( \dot{x} - q \frac{\dot{s}}{R}\right)\hat{x_{0}} + \dot{z}\hat{z_{0}} + \left( 1 + \frac{x}{R} + \frac{\dot{q}}{\dot{s}}\right)\dot{s}\hat{s_0}.
\end{equation}
It should be obvious that $\dot{s}=v_0$, the velocity of the reference particle, while $\dot{q}$ is the relative velocity between the particle of interest and the reference particle.
Lets look at the terms that Wille is missing because the $\hat{s_{0}}$ term was omitted.  If the energy spread of the beam is small then $\frac{\dot{q}}{\dot{s}} \ll 1$ and can be dropped.  For the new term associated with $\hat{x_0}$ the figure of interest is $\frac{q}{R}$ which is the particle separation over the bend radius which is typically very small in traditional accelerators, $\frac{q}{R} \ll 1$.  If you're concerned about the $\dot{s}$ in there you can perform the transformation described by 3.11 and see that the $\hat{x_0}$ term becomes $\left( x' - \frac{q}{R} \right)\dot{s}\hat{x_0}$.
