$p_i$ is its conjugate momentum [@twoform]. In particular, this means that given some closed trajectory in 6-dimensional phase space, the *sum* of the area of the projections onto the $x$, $p_x$ plane, the $y$, $p_y$ plane, and the $z$, $p_z$ plane is constant for all time, though the area of each individual projection may change.

As an example, consider a simple 1-dimensional storage ring consisting of a drift space, a focusing quad, a second drift space, and a defocusing quad. For each element Hamilton’s equations may be solved analytically. A particle which makes a single pass through this ring traces a closed trajectory in $x$, $p_x$ phase space. As the particle continues to circulate through the ring, the area of this trajectory does not change. In particular, the particle’s trajectory in phase space does not spiral in or out. The first plot in Figure 5(a) depicts the particle’s trajectory over 1000 circulations through the ring, determined by solving Hamilton’s equations analytically.

For more complicated systems, Hamilton’s equations cannot be solved analytically. Therefore, we must solve them using numeric integration schemes. In order for our simulations to be physical, it is essential that whatever scheme we use also preserves the area 