# Variational principle

1. Mar 12, 2009

### dirk_mec1

1. The problem statement, all variables and given/known data

Consider the variation principle for the space-time functional of the variables $$\eta, \phi$$

$$A( \eta, \phi) = \int \int \phi \partial _t \eta -\frac{1}{2}g \eta ^2 -\frac{1}{2} h ( \partial_x \phi ) ^2\ \mbox{d}x \mbox{d}t$$

Derive the two coupled equations for the critical points.
2. Relevant equations
$$\eta, \phi$$ are functions x and t and are the surface elevation and the potential at the surface respectively, g is gravitation constant.

Fluid depth is h and h=h(x).

3. The attempt at a solution
Do I need to find the dervatives of this double integral? To be honest I don't know how to start.

Last edited: Mar 12, 2009