Few analytical solutions are known for the Lagrangian formulation of fluid dynamics; as few, in fact, as are known for the Eulerian. Almost all these solutions describe flow free of momentum advection. The Gerstner waves, and their generalizations the Ptolemaic vortices, stand out as extraordinary exceptions. The classical investigative techniques of linearized hydrodynamic stability theory are available to both the Lagrangian and the Eulerian formulation, as are the newer techniques of phase plane analysis. The classical Stokes' problems for viscous flow near plates are solvable in both formulations; the Lagrangian self-similar solution for Blasius' approximate boundary layer dynamics is intriguingly more complicated than the Eulerian. The general solvability of the Lagrangian formulation of inviscid incompressible fluid dynamics appears, to an applied mathematician, to depend upon the choice of dependent variables. The classical solvability of the viscous problem comes exasperatingly close to being provable in the large, but in the end remains an open question for pure mathematicians.