each magnetic field component, and divided by the total number of data points.\

[|l|l|l|l|]{}

C-Shaped LGB–7 Terms & $B_x$ $(T)$ & $B_y$ $(T)$ & $B_z$ $(T)$

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Maximum Residual Magnitude & 0.0 & 1.4[$\times 10^{-1}$]{} & 4.2[$\times 10^{-1}$]{}

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Sum of Residual Magnitudes & 0.0 & 8.6[$\times 10^{2}$]{} & 1.6[$\times 10^{2}$]{}

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Average Residual (23,776 data points) & 0.0 & 3.6[$\times 10^{-2}$]{} & 6.8[$\times 10^{-3}$]{}

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Canted Solenoid LGB–9 Terms & $B_x$ $(T)$ & $B_y$ $(T)$ & $B_z$ $(T)$

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Maximum Residual Magnitude & 8.1[$\times 10^{-2}$]{} & 7.9[$\times 10^{-2}$]{} & 3.4[$\times 10^{-2}$]{}

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Sum of Residual Magnitudes & 1.78[$\times 10^{2}$]{} & 2.0[$\times 10^{2}$]{} & 1.3[$\times 10^{2}$]{}

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Average Residual (50,900 data points) & 3.0[$\times 10^{-3}$]{} & 3.3[$\times 10^{-3}$]{} & 2.1[$\times 10^{-3}$]{}

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\

Building a Symplectic Integrator
================================

Symplectic Integration Overview
-------------------------------

In this section, we give a brief introduction to symplectic integration and explain why it is a necessary procedure in particle tracking simulations. We begin with a fact from Hamiltonian mechanics: Hamiltonian flows are symplectic, meaning they preserve the area associated with the following two-form:

$\omega = \displaystyle\sum_{i = 1}^{3} dp_i \wedge dq_i$,

in which $q_i$ is a generalized coordinate and 