In which sense is the configuration variable of a classical spin SU(2)? I can view a classical spin as a unit vector in S^2 (2dim sphere), but it seems it is really given by a matrix U in SU(2). The Hopf map $H:SU(2)\rightarrow S^2$ given by $H(U)=U\sigma_3U^{\dagger}$ whose image can be identified with an element in $S^2$ gives what I imagined to be this classical spin.
Since with a magnetic field $B$ the interaction is just $H(U) \cdot B$ there would be no problem on just considering $S \in S^2$ as a configuration variable, but I read the following:
A classical particle of mass $m$, with position $x$ and spin $S$ moving on a fixed external magnetic field $B$ can be described by the Lagrangian function on the tangent bundle of the configuration space $\mathbb{R} \times SU(2)$ given as
$L=\frac{1}{2}\dot{x}^2+i\lambda Tr(\sigma_3U^{\dagger}\dot{U})+\mu Tr(H(U)\dot B)$
So the second term is explicitly in terms of $U$.
Thank you
EDIT: EDIT: I got this from "Gauge symmetries and fiber bundles". Balachandran et al.
