Neumann boundary conditions in calculus of variations

[ShayanJ]

Main Question or Discussion Point

In calculus of variations, extremizing functionals is usually done with Dirichlet boundary conditions. But how will the calculations go on if Neumann boundary conditions are given? Can someone give a reference where this is discussed thoroughly? I searched but found nothing!
Thanks

 

[Orodruin]

What usually results in Neumann boundary conditions is variation with free boundaries, i.e., when no boundary condition is given. The boundary condition then results from the extremisation.

 

[ShayanJ]

Can you explain in more details or point to a reference that does?

 

[Orodruin]

When deriving the EL equations, you do a partial integration. The usual assumption that the function is fixed at the end-points makes the boundary terms disappear. If you do not make this assumption, you will need the boundary term to vanish for all variations in order for the variation to be zero. This requirement leads to $\partial\mathcal L/\partial \dot y = 0$, where $y$ is the function you are deriving the EL equation for.

Edit: Naturally, the condition is only valid at the free boundary.

 

[Orodruin]

This should be explained in any introduction to variational calculus. Just a reference I happen to have at hand is page 782 of Riley, Hobson, Bence, 3rd ed. Should be in Arfken and Boas as well, but my copies are at work and I am not.

 

[ShayanJ]

What you explained is surely part of what I asked for, but that doesn't seem to be the whole story. Because now we've covered two cases: 1)When the function itself is given at the boundaries. 2) When no boundary condition is given.
But another condition, which is what I mean by Neumann boundary conditions and I don't see how it is equivalent to not giving any boundary conditions at all, is specifying the normal derivative of the function at the boundaries. This is surely different from what you say because when it is specified that the variation should be done in a way that $\dot y|_{\partial D} =g(x)$, there is no guarantee that $\partial \mathcal L/ \partial \dot y=0$ will be the same as the specified boundary condition!

 

[Orodruin]

Indeed there is no such requirement. But Neumann conditions in physics will generally appear because of this procedure, not by imposition.

 

[ShayanJ]

Maybe its rare in physics, but surely its possible to do it as a mathematical exercise. Then what? Should I ask this in the math section?

 

[Orodruin]

I am not sure you can consistently do this (for an integrand with first order derivatives). I have not considered it that much though. Obtaining the EL ewuations requires you to do the integration by parts. If you do not have the Dirichlet condition, having zero variation will result in the free boundary condition and you will be left with too many boundary conditions. You might be able to impose it as an additional constraint and use Lagrange multipliers to figure something out, but again, I did not think too much about it (and I just woke up ;)).