(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider the variation principle for the space-time functional of the variables [tex] \eta, \phi [/tex]

[tex] 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 [/tex]

Derive the two coupled equations for the critical points.

2. Relevant equations

[tex] \eta, \phi [/tex] 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.

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# Variational principle

Can you offer guidance or do you also need help?

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