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In summary, the condition to achieve Neumann boundary conditions is to specify the normal derivative of the function at the boundaries.

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

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

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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!

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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.

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.

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.

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.

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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