Neumann boundary conditions in calculus of variations

In summary, the condition to achieve Neumann boundary conditions is to specify the normal derivative of the function at the boundaries.
  • #1
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!
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  • #2
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.
  • #3
Can you explain in more details or point to a reference that does?
  • #4
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.
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  • #5
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.
  • #6
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!
  • #7
Indeed there is no such requirement. But Neumann conditions in physics will generally appear because of this procedure, not by imposition.
  • #8
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?
  • #9
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 ;)).

1. What are Neumann boundary conditions in calculus of variations?

Neumann boundary conditions refer to a type of boundary condition in the calculus of variations, which is a branch of mathematics that deals with finding optimal functions. These conditions specify the behavior of a function at the boundaries of a given domain, by setting the value of the derivative of the function at those boundaries.

2. How are Neumann boundary conditions different from Dirichlet boundary conditions?

The main difference between Neumann and Dirichlet boundary conditions is that Neumann conditions specify the behavior of the function's derivative at the boundary, while Dirichlet conditions specify the value of the function itself at the boundary. In other words, Neumann conditions involve the first derivative of the function, while Dirichlet conditions involve the function itself.

3. Why are Neumann boundary conditions important in the calculus of variations?

Neumann boundary conditions are important because they allow us to find the optimal function that satisfies both the boundary conditions and the Euler-Lagrange equation, which is the fundamental equation in the calculus of variations. These conditions also play a crucial role in solving various physical and engineering problems, such as finding the shape of a soap film or the path of a light ray.

4. Can Neumann boundary conditions be applied to any type of function?

Yes, Neumann boundary conditions can be applied to any type of function, as long as it satisfies the Euler-Lagrange equation. However, it is important to note that Neumann conditions are more commonly used for functions that describe physical phenomena, such as temperature, electric potential, or fluid flow.

5. How are Neumann boundary conditions incorporated into the calculus of variations?

In the calculus of variations, Neumann boundary conditions are incorporated by adding them as constraints to the functional that is being optimized. This means that the optimal function must not only satisfy the Euler-Lagrange equation, but also the specified behavior at the boundaries. These conditions can be solved using various techniques, such as the method of lagrange multipliers or the method of finite differences.

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