A Neumann boundary conditions in calculus of variations

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.

 

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.

 

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.

 

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

 

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 ;)).