# Calculus of variations question from Darcogona

I asked the question in the next forum, the 4th post, hopefully someone can help me with this here or there: https://nrich.maths.org/discus/messages/7601/151442.html?1310911861

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Please post the question here in full instead of merely linking to it. –  Ｊ. Ｍ. Jul 17 '11 at 14:31

Yes, this is what happens when authors mix partial derivatives and variable changes indiscriminately without making any effort to indicate what's being varied and what's being held fixed in a given context.

$f$ is a function of three variables, and they can all be varied independently, so there are three partial derivatives of $f$ with respect to its three arguments, the other two being held fixed. $\xi$ and $\eta$ are functions of two variables, so they each have two partial derivatives with respect to their two arguments, the other being held fixed. So far, so clear.

The second argument of both $\xi$ and $\eta$ is $\epsilon$, but the first arguments are different, so the partial derivatives of $\xi$ and $\eta$ with respect to $\epsilon$ are different derivatives in different coordinate systems. (By this I mean not just that obviously the derivatives of two different functions are different, but that two different differential operators are being applied to these two functions, both denoted by the same notational device (a subscript $\epsilon$)). When the author then writes the integrand as $g(\epsilon)$ and differentiates it with respect to $\epsilon$ by adding a prime, it's not immediately obvious which of these two derivatives is intended. However, the "change of variables $y=\xi(x,\epsilon)$" suggests that $y$ and $\epsilon$ are now to be regarded as the independent variables, and indeed the calculation of $g'$ makes sense if you interpret it that way. For your calculation, you interpreted it not just the other way, but both ways -- your expression contains both $\partial\xi/\partial\epsilon$ and $\eta_\epsilon$ -- but since these arise from applying two different differential operators, at most one of them could be the intended result of the differentiation with respect to $\epsilon$. You'll get the right result from your expression if you set $\partial\xi/\partial\epsilon$ to zero, which indeed it is if $y$ and $\epsilon$ are the independent variables, since $\xi=y$.

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This mistake is made surprisingly often. I often see people define a coordinate function, say $r$, and proceed to take the $\partial/\partial r$ derivative without ever bothering to define the rest of the coordinate system... I was at a conference last year and heard someone call this "The Second Fundamental Mistake of Multivariable Calculus". I didn't inquire to what the first is. –  Willie Wong Jul 21 '11 at 15:08

Hilbert was the first to give good conditions for the Euler Lagrange equations to give a stationary solution. Within a convex area and a positive thrice differentiable Lagrangian the solutions are composed of a countable collection of sections that either go along the boundary or satisfy the Euler Lagrange equations in the interior. However Lavrentiev in 1926 showed that there are circumstances where there is no optimum solution but one can be approached arbitrarily closely by increasing numbers of sections. Here a zig zag path gives a better solution than any smooth path and increasing the number of sections improves the solution. The difficulty with this reasoning is the assumption that the minimizing function u must have two derivatives. Riemann argued that the existence of a smooth minimizing function was assured by the connection with the physical problem: membranes do indeed assume configurations with minimal potential energy. Riemann named this idea the Dirichlet principle in honor of his teacher Dirichlet. However Weierstrass gave an example of a variational problem with no solution: minimize.

. f is a function of three variables, and they can all be varied independently, so there are three partial derivatives of f with respect to its three arguments, the other two being held fixed. ξ and η are functions of two variables, so they each have two partial derivatives with respect to their two arguments, the other being held fixed. So far, so clear.

This corresponds to an external force density f(x,y) in D, an external force g(s) on the boundary C, and elastic forces with modulus σ(s) acting on C. The function that minimizes the potential energy with no restriction on its boundary values will be denoted by u. Provided that f and g are continuous, regularity theory implies that the minimizing function u will have two derivatives. In taking the first variation, no boundary condition need be imposed on the increment v. Note that this boundary condition is a consequence of the minimizing property of u: it is not imposed beforehand. Such conditions are called natural boundary conditions The functions p(x) and r(x) are required to be everywhere positive and bounded away from zero. The primary variational problem is to minimize the ratio Q/R among all φ satisfying the endpoint conditions. It is shown below that the Euler-Lagrange equation for the minimizing u is by the variational problem also applies to more general boundary conditions. Instead of requiring that φ vanish at the endpoints, we may not impose any condition at the endpoints, and set where a1 and a2 are arbitrary as positive convex planes. Fermat's principle states that light takes a path that (locally) minimizes the optical length between its endpoints. If the x-coordinate is chosen as the parameter along the path, and y = f(x) along the path, then the optical length .

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