Generalized Hopf Fibration and Geometric SO(3) Reduction of the 4DOF Harmonic Oscillator

It is shown that the generalized Hopf map ℍ×ℍℍ×ℍ → ℍ×ℝ×ℝℍ×ℝ×ℝquaternion formulation can be interpreted as an SO(3) orbit map for a symplectic SO(3) action. As a consequence the generalized Hopf fibration S7S7 → S4S4 appears in the SO(3) geometric symplectic reduction of the 4DOF isotropic harmonic oscillator. Furthermore it is shown how the Hopf fibration and associated twistor fibration play a role in the geometry of the Kepler problem and the rigid body problem.