The dynamic aperture is the stability region of phase space in a circular accelerator.
In the case of protons or heavy ion accelerators, (or synchrotrons, or storage rings), there is minimal radiation, and hence the dynamics is symplectic.
A scaling law for Dynamic aperture vs. number of turns has been proposed by Giovannozzi.
The basic method for computing dynamic aperture involves the use of a tracking code.
A model of the ring is built within the code that includes an integration routine for each magnetic element.