Miles: a new nonperturbative formalism to calculate the invariant spin field in circular accelerators

We describe a new nonperturbative algorithm called Miles to calculate the invariant spin field in circular accelerators. It is a map-based algorithm, based exclusively on the field-theoretic transformation of a vector field under the flow in the orbital phase-space. We employ the method to derive the exact analytical solution for a model of a planar ring with one Siberian Snake and a single resonance driving term. This model is in some sense the classic problem in the field. The solution contains new types of mathematical functions, which we call “sine-factorial” and “sine-Bessel” functions. We also display the exact analytical solutions for several other storage ring models, including rings with two or more Snakes. We compare our method against other formalisms, including numerical tracking programs.