  With this paper we extend our studies [1] on polarized beams by distilling
tools from the theory of principal bundles. Four major theorems are presented,
one which ties invariant fields with the notion of normal form, one which
allows one to compare different invariant fields, and two that relate the
existence of invariant fields to the existence of certain invariant sets and
relations between them. We then apply the theory to the dynamics of spin-1/2
and spin-1 particles and their density matrices describing statistically the
particle-spin content of bunches. Our approach thus unifies the spin-vector
dynamics from the T-BMT equation with the spin-tensor dynamics and other
dynamics. This unifying aspect of our approach relates the examples elegantly
and uncovers relations between the various underlying dynamical systems in a
transparent way.
