Quaternions, Spinors and the Hopf Fibration: Hidden Variables in Classical Mechanics.

Rotations in 3 dimensional space are equally described by the SO(3) and SU(2) groups. These groups are isomorphic as they are both generated by the unit quaternion, thereby generating the same kinematics in 3D space. The Hopf Fibration describes the projection between the hypersphere $\mathbb{S}^3$ of the quaternion in 4D space, and the unit sphere $\mathbb{S}^2$ in 3D space. Great circles in $\mathbb{S}^3$ are mapped to points in $\mathbb{S}^2$ via the 6 Hopf maps and their respective stereographic projections. The higher and lower dimensional spaces are connected via the $\mathbb{S}^1$ fibre bundle which consists of the global, geometric and dynamics phases. The global phase is quantized in integer multiples of $2\pi$ and presents itself as a natural hidden variable of Classical Mechanics.