The Hamiltonian structure of surface gravity waves on a potential flow was discovered by Vladimir E. Zakharov in 1968, and rediscovered independently by Bert Broer and John Miles:
where the surface elevation  and surface potential  — which is the potential  at the free surface  — are the canonical variables. The Hamiltonian  is the sum of the kinetic and potential energy of the fluid:
The additional constraint is that the flow in the fluid domain has to satisfy Laplace's equation with appropriate boundary condition at the bottom  and that the potential at the free surface  is equal to :