  The unit sphere $\mathbb S^3$ can be identified with the unitary group SU(2).
Under this identification the unit sphere can be considered as a
non-commutative Lie group. The commutation relations for the vector fields of
the corresponding Lie algebra define a 2-step sub-Riemannian manifold. We study
sub-Riemannian geodesics on this sub-Riemannian manifold making use of the
Hamiltonian formalism and solving the corresponding Hamiltonian system.
