Hamiltonian Formulation
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 η and surface potential φ — which is the potential Φ at the free surface z=η(x,t) — 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 z=-h(x) and that the potential at the free surface z=η is equal to φ:
Read more about this topic: Luke's Variational Principle
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