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Is there a physical example of a field that would have the following Lagrangian density $$ L= \sqrt{1+\phi_x^2 +\phi_y^2+\phi_z^2} $$ where the subscripts denote partial derivatives and $\phi$ is a scalar field?

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Can I ask where you found this Lagrangian? It looks interesting. – Kasper Meerts Jun 27 '11 at 17:59
By the way, in physics nomenclature these types of Lagrangian densities are often called (static) non-linear sigma models. Some people would also refer to the particular form that you wrote down as one that is of "Born-Infeld type". – Willie Wong Jun 28 '11 at 1:58

This looks a lot like soap film statics, but with an extra dimension. Consider a soap film glued to a ring. The film is described by a function $z = \phi(x,y)$, with $z$ the height of the film above the xy-plane. We want to minimise the potential energy of the film, which means to a good approximation minimising the surface area. The total area of the film is given by

$ A = \int \sqrt{1 + \phi_x^2 + \phi_y^2}$

where the integration is over the entire xy-plane. This functional looks very similar to your Lagrangian density. The solution can be derived by applying the Euler-Lagrange equations and the solution is confusingly also called Lagrange's equation

$ (1 + \phi_y)^2 \phi_{xx} + 2 \phi_x \phi_y \phi_{xy} + (1 + \phi_x)^2 \phi_{yy} = 0 $

To give you an idea of what the solutions look like, if the field doesn't change too rapidly we can make some approximations ($\phi_x \approx \phi_y \approx \phi_{xy} \approx 0$) and get Laplace's equation

$ \phi_{xx} + \phi_{yy} = \Delta \phi = 0$

This also provides an interesting interpretation for solutions to the Laplace equation, as they are approximate minimal-area surfaces.

Most of this carries over three dimensions but the minimal-surface interpretation isn't so clear anymore.

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The Euler Lagrange equation for what you wrote down as the action is not $\triangle \phi = 0$. It should be instead Lagrange's equation. The higher dimensional generalisation does not have the solution as a harmonic field either. – Willie Wong Jun 27 '11 at 16:51
Also, the Lorentzian version of the equation (with density $\sqrt{1 - (\partial_t\phi)^2 + (\partial_x\phi)^2 + (\partial_y\phi)^2}$ ) is sometimes called the "relativistic membrane equation"; you can see this article of Hoppe for a discussion and further info. – Willie Wong Jun 27 '11 at 17:02
@Willie Wong: Rats, I forgot to take the derivative of the numerator. I'll edit my answer. – Kasper Meerts Jun 27 '11 at 17:50
@Kasper I encountered this Lagrangian in the book "Emmy Noether's Wonderful Theorem" by Neuenschwander. – Noah Jun 27 '11 at 21:39
Okay, +1 now that you fixed the error. :-) Note that the minimal surface interpretation is still valid: it is just not a surface any more, but a hypersurface in $\mathbb{R}^4$ (so you extremize 3-dimensional volume). – Willie Wong Jun 28 '11 at 1:52

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