Luke's Variational Principle - Hamiltonian Formulation

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:

$\begin{align} \rho\, \frac{\partial\eta}{\partial t}\, &=\, +\, \frac{\delta\mathcal{H}}{\delta\varphi},\\ \rho\, \frac{\partial\varphi}{\partial t}\, &=\, -\, \frac{\delta\mathcal{H}}{\delta\eta}, \end{align}$

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:

$\mathcal{H}\, =\, \iint \left\{ \int_{-h(\boldsymbol{x})}^{\eta(\boldsymbol{x},t)} \frac12\, \rho\, \left[ \left| \boldsymbol{\nabla}\Phi \right|^2\, +\, \left( \frac{\partial\Phi}{\partial z} \right)^2 \right]\, \text{d}z\, +\, \frac12\, \rho\, g\, \eta^2 \right\}\; \text{d}\boldsymbol{x}.$

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:

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“In necessary things, unity; in disputed things, liberty; in all things, charity.”
—Variously Ascribed.

The formulation was used as a motto by the English Nonconformist clergyman Richard Baxter (1615-1691)