A note regarding ‘On Hamilton's principle for surface waves’

The canonical form of the equations for the free-surface elevation and potential of an irrotational fluid is more than a coincidence. The elevation is a ‘generalized coordinate’ field sufficient to define the system Lagrangian without explicit reference to the motion of the fluid interior. The Lagrangian and the associated field equations are complete and self-contained in the two-dimensional surface co-ordinates, but non-local (integro-differential) in form; the canonical equations derived by Miles are just the Hamiltonian counterparts.