The case of spin

We will now describe the normal form in FPP for the spin orbital map when the spin is a spectator. Again it is simplest to describe it in words first. In the commutative normal form of the spin orbital map, the orbital part moves on its usual circles with an amplitude dependent tune. Nothing is new there. The spin maps are all rotation along an arbitrary axis, we pick the vertical $y$-axis. The angle of the spin rotation depends only of the orbital motion, i.e., on the radii of the circles-- on the action variables!

This is so trivial that a child can understand it. First of all, the $y$-axis is certainly an invariant. The spin angle being a function of the actions $J_i$ is certainly a well defined tune since it is constant along the trajectory. It is the so-called spin tune. Finally since it is a commutative normal form, we can define easily a spin phase advance and a spin adiabatic invariant. But we are going too far here.

The normalized motion in the presence of spin is like the small and large arms of an old fashion watch. The small arm marks the time (orbital) and the little arm moves at a faster rate totally related to the hourly rate of the small arm. The amplitude dependence would be like saying that making the clock smaller/larger in size would result in a clock not well adjusted. Actually that would probably be the case if one scales the size of the clock without retuning it.

In the rest of the text we may use the locution ``normal norm'' to describe the Taylor map algorithm of FPP. The reader will perhaps excuse the abuse of language. Always one should remember that the tracking code is the final arbiter on the validity of an algorithmically computed normal form.