  We investigate geometric evolution equations for Legendrian curves in the
3-sphere which are invariant under the action of the unitary group ${\rm
U}(2)$. We define a natural symplectic structure on the space of Legendrian
loops and show that the modified Korteweg-de Vries equation, along with its
associated hierarchy, are realized as curvature evolutions induced by a
sequence of Hamiltonian flows. For the flow among these that induces the mKdV
equation, we investigate the geometry of solutions which evolve by rigid
motions in ${\rm U}(2)$. Generalizations of our results to higher-order
evolutions and curves in similar geometries are also discussed.
