Fluid mechanics Water waves and free surfaces Theorem

Luke's variational principle

Builds on Water waves Calculus of variations Velocity potential

Statement

Let $g > 0$ , $h > 0$ , let $D \subset \R^d$ be a bounded horizontal domain and $[t_0, t_1]$ a time interval. For smooth functions $\eta(x, t) > -h$ on $D \times [t_0, t_1]$ and $\phi(x, z, t)$ on $\{(x, z, t) : -h \le z \le \eta(x, t)\}$ , define the action $\mathcal{L}(\phi, \eta) = -\int_{t_0}^{t_1}\int_D \int_{-h}^{\eta(x, t)} \left(\partial_t\phi + \tfrac{1}{2}\abs{\nabla\phi}^2 + gz\right) dz\,dx\,dt,$ where $\nabla$ is the gradient in $(x, z)$ . Consider variations of $\phi$ and $\eta$ that vanish at $t = t_0$ , at $t = t_1$ and on the lateral boundary $\partial D \times \R$ . Then $(\phi, \eta)$ is a stationary point of $\mathcal{L}$ for all such variations if and only if the following hold. These are the equations of the water wave problem with flat bottom at $z = -h$ .

$\Delta\phi = 0$ for $-h < z < \eta$ .
$\partial_z\phi = 0$ at $z = -h$ .
$\partial_t\eta + \nabla_x\phi\cdot\nabla_x\eta = \partial_z\phi$ at $z = \eta$ .
$\partial_t\phi + \tfrac{1}{2}\abs{\nabla\phi}^2 + g\eta = 0$ at $z = \eta$ .

Remarks

By the Bernoulli equation the integrand is minus the pressure divided by the density. So the action is the integral of the pressure over the fluid and over time, divided by the density. Varying $\eta$ changes only the upper limit of the inner integral. Its first variation is the left side of the dynamic condition $\partial_t\phi + \tfrac{1}{2}\abs{\nabla\phi}^2 + g\eta = 0$ on the surface. Varying $\phi$ and integrating by parts in $(x, z, t)$ gives a volume term $\Delta\phi$ , a bottom term $\partial_z\phi$ and a surface term. The surface term is the kinematic condition, and each of the three terms must vanish separately. J. C. Luke published the principle in 1967. With $\eta$ and the surface value of $\phi$ as variables, the principle leads to Zakharov's Hamiltonian formulation of water waves.

Going deeper

For a student

The principle is a tool of the calculus of variations applied to a free boundary problem. It is the starting point of Whitham's averaged variational method for slowly varying wave trains. Many approximate models also come from it, by restricting $\phi$ to a simple form in $z$ before varying. Such restrictions preserve the variational structure, and so conservation laws of energy and momentum carry over to the approximate equations.

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Tag	`5Z141B47T0`
URI	`m2://5Z141B47T0`
Kind	statement, theorem
Writing	W0 stub
MSC	76B15, 76M30
Status	This copy of the site has no build data. Run `m2 status 5Z141B47T0`.
About	Water waves
Uses	Water waves, Calculus of variations, Velocity potential
Same as	Wikidata Q6701750 Wikipedia Luke's variational principle
Changed	3 November 2026

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