I Minimal Surface shape with gravity

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1. Apr 6, 2017

DuckAmuck

Minimal surfaces are sort of the "shortest path" but in terms of surface shapes.
So I figured I could characterize the shape of a hammock by adding the influence of gravity, much like you can get the shape of a catenary cable (y=cosh(x)).

The equation of motion I get from the Lagrangian is:
$$z_x^2 + z_y^2 + 1 = z ( z_{xx} (z_y^2 + 1) + z_{yy} (z_x^2 +1) - 2 z_x z_y z_{xy} )$$
where z is the height of a point on the surface mapped to (x,y).

Of course, this is likely to have non-unique solutions just like other minimal surfaces.
One of the solutions I found is a cone:
$$z = \sqrt{x^2 + y^2}$$
What does *not* work as a solution is a "2-d catenary", which is what I initially suspected as solution
$$z = cosh(x)cosh(y)$$
Anyone else attempt this kind of problem? What were your findings? I'm basically just plugging things into the equation of motion and seeing if they work.

2. Apr 7, 2017

Nidum

You can do these types of problem for sheets of material with real physical properties using standard engineering analysis methods .

No initially flat sheet with edge restraints can distort into any other shape without some stretching of the material .

3. Apr 12, 2017

DuckAmuck

Would you mind linking me to a good source for this kind of engineering analysis methods?

4. Apr 12, 2017

Nidum

Last edited: Apr 12, 2017