To me, one of the most beautiful pieces of mathematics is the Euler-Lagrange equations. The idea is this: you are seeking to minimize a *functional, *which is a function that eats functions and gives numbers. You won’t lose much by assuming that this functional is an integral. In particular, we will assume we are dealing with minimizing

where , and (as usual) *Du* is the vector of partial derivatives of *u*. For example, perhaps which would be minimized by choosing *u(x)* identically equal to zero. Or maybe we want to minimize , subject to the requirement that *u(0) = u(1) = 1*. Note that our desired “solution” will be a function on, in this case, the unit interval.

One could spend a while talking about classic problems in the calculus of variations (which the study of these problems is called), but I would like to move right to how to *solve* these problems. What we will do is extract a partial differential equation which will be zero at the minimizer (though it might be zero in other places too). The strategy is as follows: assuming we have a minimizer, “poking” the function in any direction will make *L[u]* larger. This is similar to the calculus argument that a relative extrema will have derivative zero. But in (single variable) calculus, we may “perturb” a point by moving it to the right or left on the real line. How do we “perturb” a function?

Well, we add a “small” function to it. Namely, we will suppose that *u* is a minimizer with the necessary boundary data (one typically specifies boundary data in these sorts of problems, to guarantee uniqueness), and let *v* be a smooth, compactly supported function. Then, for any real number *t*, we have , and that *u+tv* has the desired boundary data. So now we have a function of a single real variable, call it *f(t)*, and have deduced that it has a minimum when *t = 0*.

Specifically, we have

In order to differentiate, the notation gets sort of bad. In practice, just remembering that you should differentiate now is pretty good, but let’s forge on. We notice that *F *is a map from a big vector space to the reals. Namely, , since *x* and *Du* are vectors. I will denote the derivative of *F* with respect to its first *n* variables by , and the derivative of *F* with respect to *u* by . The *p* in the first derivative is convention, and notice that is a number, not a vector. Also, I won’t need the derivative with respect to the last *n* variables, but if I did, I’d write $D_xF$.

Now then, differentiating with respect to *t*, we get

so

Now if you’re really sharp with integrating by parts (and remember that *v* has compact support, and so vanishes on the boundary of ), you would probably know that the left hand summand may be transformed (and *f'(0)* swapped with *0*) to get

which can be rewritten as

*all*smooth functions

*v*. Hence if the big ugly guy is nonzero anywhere in , we choose a

*v*that is supported right there, and break the equality. This sentence can be fleshed out into a full argument that the big ugly guy (which maybe I’ll call the

*Euler-Lagrange equation*instead) must vanish at any critical point of the functional. Once more for emphasis:

*u*is a critical point of the functional

*L*, then

*u*satisfies the Euler-Lagrange equation

$latex D_uF(Du,u,x)-div_x(D_pF(Du,u,x)) = 0.$

I’ve had enough integral signs for one day, but at least I got to post another nice .gif I had lying around!

[…] to some boundary conditions. Longtime reader(s) will recall the Euler-Lagrange equationswhich can The *only* function that minimizes length, with u(0) = a, u(1) = […]