Relevant Wikipedia Articles:
Local Lagrangian Densities
Extremize over , where :
where is a real-valued Lagrangian density and is the measure for . Specifically, we wish to solve for this extremization in terms of the functional derivative
where the function is arbitrary up being zero on the boundary (denoted ) of , and , , and are differentiable. Let us expand the induced perturbation to first order in about :
for scalar , vector , and unit vector normal to .
In our case,
The only way to guarantee that this derivative is zero for all permissible is for the Euler-Lagrange equation to hold:
Non-Local Lagrangian Densities
Consider now a situation in which the Lagrangian density depends on the joint behavior of at disjoint points indexed by , such that the relationship between these points is captured by . That is, we seek to extremize depends on non-local properties of the function , and determines any specific constraints or symmetries of the points.
Using the same prescription to vary as above, let
This time, however, rather than fixing the value of on , we will assume either a periodic structure for , or a Lagrangian density that does not depend on for all , in order to guarantee
Let us determine the perturbation to first order in by considering perturbations of and at each separately:
We again integrate by parts, discarding the boundary term by previous assumption (Eq. (3)), and set at the desired extremum. We obtain
Let us be mathematically explicit about how this equation must hold for all . Specifically, consider perturbations of the form where is the Dirac delta, for any . It follows that
A Sanity Check
Consider the problem We know that translations of minimize this objective, though we have not excluded other possibilities.
To solve this problem with variational calculus, we redefine the Lagrangian density as and integrate over the manifold
Applying Eq. (4), we have
Indeed, translations of solve this equation, as hoped. Without ruling out other solutions, Eq. (4) at least passes this test.