s}{2}: H_1:)\exp(-\Delta s :H_2:) \\
\exp(-\frac{\Delta s}{2}: H_1:) + O(\Delta s^3).\end{gathered}$$

This is the final form of our simple symplectic integrator–all that is left to do is evaluate the composition of Lie exponential operators, which requires a particular Hamiltonian. It is useful to turn an exponential of a sum into a product of exponentials because it is typically easier to evaluate Lie exponential operators for $H_1$ and $H_2$ and compose them than it is to evaluate a single exponential operator for $H$. In fact, one method of evaluating a Lie exponential operator for an arbitrary Hamiltonian is analytically solving Hamilton’s equations; according to equation (5) the two are one in the same. Thus, what we are effectively doing is decomposing the Hamiltonian into different terms, analytically solving Hamilton’s equations for each term, and composing the resulting maps as in equation (6).\

Modifications to Existing Bmad Symplectic Integration Software
--------------------------------------------------------------

Before this project, there existed a Bmad symplectic integrator program specifically designed for wiggler magnets. In particular, looking back to equation (3), the existing integrator assumed $\phi_x$ and $\phi_y$ to be zero for every term, and did not employ a constant offset as we do in our model. Since we could 