associated with the aforementioned two-form. Such a scheme is called a symplectic integrator. If we do not use a symplectic integrator, then our tracking simulations do not accurately depict the stability of a particle beam in a storage ring. If the $x$ coordinate of a simulated particle grew without bound so that the particle was lost from the ring, it would be impossible to know if this were due to a numerical artifact or the design of the ring itself.

To demonstrate the adverse effects of not using a symplectic integrator, we refer back to our simple example. Figure 5(b) depicts the trajectory of a simulated particle in phase space for the case in which we solved Hamilton’s equations numerically using Euler’s method, which is not symplectic. Evidently, the particle’s trajectory spirals outward. Since we are tracking through the same ring as before, this behavior must be an artifact of our integration scheme. Figure 5(c) depicts the trajectory of simulated particle in which we solved Hamilton’s equations numerically using a Runga-Kutta scheme. While at first glance it may appear that this scheme is symplectic and preserves the area of the ellipse in phase space, zooming-in (Figure 5(d)) demonstrates that the 