Variational derivative of mean curvature

[Hologram0110]

Hello Physics Forums,

I have a simple parametric surface in R3 <x,y,z(x,y)>. I've calculated the the usual mean curvature:

H= ((1+hx^2)hyy-2hxhyhxy+(1+hy^2)hxx)/(1+hx^2+hy^2)^3/2

I needed to take the variational derivative of this expression. Since it has second order spatial derivatives the equation is:

d/dh - d/dx d/dhx - d/dy d/dhy + d^2/dx^2 d/dhxx + d^2/dxdy d/hxy +d^2/dyy d/hyy

To simplify the math I grouped the terms by d/dx and d/dy and I found that :
-dH/dhx + d/dx dH/dhxx + 1/2 d/dy dH/dhxy = 0
and
-dH/dhy + d/dy dH/dhyy + 1/2 d/dx dH/dhxy = 0

I've encountered this in my work by brute forcing it with Mathematica. However, it seems like there might be a more fundamental or elegant way of getting the result. Is there some well known reason for this? I'm an engineer working in material science. I've never taken a course on variational calculus or differntial geometry so please forgie me if this is obvious.

 

[jvicens]

Hello,

Thank you for your post. It is always exciting to see someone exploring the applications of mathematics in their field of work.

To answer your question, the reason for the result you found is related to the fundamental theorem of calculus of variations. This theorem states that for a functional F defined on a space of functions, the critical points of F correspond to the solutions of the Euler-Lagrange equations. In your case, the functional is the mean curvature H and the critical points are the parametric surfaces that satisfy your equation.

By taking the variational derivative of H, you are essentially solving the Euler-Lagrange equations for the functional H. This is why you are getting the result you found, which is a set of equations that relate the first and second order derivatives of H. These equations are known as the Euler-Lagrange equations for H.

In summary, the result you found is not a coincidence, but rather a consequence of the fundamental theorem of calculus of variations. I hope this helps to clarify things for you. Keep up the good work in your research!