the normalized vector potential for the constant term in the $\mathbf{B}$-field series and substituted this into our previous evaluations of the Lie exponentials. Thus for the $b_y$ term we have that

$\exp(:\pm\displaystyle\int_{0}^{y} a_y dy':)  :
\begin{cases}
p_y \rightarrow p_y      \\
p_x \rightarrow p_x 
\end{cases}$,

$\exp(:\Delta s a_z:)  :
\begin{cases}
p_y \rightarrow p_y     \\
p_x \rightarrow p_x - \Delta s\frac{q}{P_0 c}b_y 
\end{cases}$.\
We note that in addition to carrying out the aforementioned symplectic integration scheme, the Bmad symplectic integrator also calculates the linear transfer map (that is, the Jacobian) associated with each transformation in equation (7). From the linear transfer maps we can calculate linear optics parameters, in particular the $\beta$-function, dispersion, and phase advance, at arbitrary locations within a magnet component. Only the linear transfer maps corresponding to Lie exponentials involving the constant term in the vector potential series, namely, those Lie exponentials we just evaluated, required any changes. The modifications to the code were straightforward. Taking the Jacobian of the previous Lie exponentials yields the identity matrix in all cases.

Particle Tracking
=================

Methods
-------

After making the proper changes to the Bmad symplectic integrator, we were able to begin particle tracking simulations. We began with a simulation lattice 