  A didactic description of charge and spin equilibrium currents on mesoscopic
rings in the presence of Spin-Orbit interaction is presented. Emphasis is made
on the non trivial construction of the correct Hamiltonian in polar
coordinates, the calculation of eigenvalues and eigenfunctions and the
symmetries of the ground state properties. Spin currents are derived following
an intuitive definition and then a more thorough derivation is built upon the
canonical Lagrangian formulation that emphasizes the SU(2) gauge structure of
the transport problem of spin 1/2 fermions in spin-orbit active media. The
quantization conditions that follow from the constraint of single-valued Pauli
spinors are also discussed. The targeted students are those of a graduate
Condensed Matter Physics course.
