the Euler-Lagrange equation
/ OY-ler lah-GRAHNZH /
Light takes the quickest path; a soap film takes the shape of least area; a hanging chain settles into the curve of lowest potential energy. Nature is full of quantities that get minimised. The Euler-Lagrange equation is the universal translator between such a minimisation principle and a differential equation: it tells you that whatever function minimises an energy integral must satisfy a specific PDE, and conversely many PDEs are secretly the minimisers of some hidden energy.
Suppose you want to minimise an energy of the form E[u] = the integral over the domain of a Lagrangian L(x, u, grad u). At a minimiser u, nudge it by a small test function: replace u by u + epsilon v (with v vanishing on the boundary) and demand that the derivative of E with respect to epsilon vanish at epsilon = 0 — the first variation is zero, the calculus-of-variations version of setting a derivative to zero at a minimum. Expanding and integrating by parts to peel the derivatives off v, you are left with the integral of v times (something) = 0 for every v. Since v is arbitrary, that something must vanish pointwise — and that something is the Euler-Lagrange equation: - div(the gradient of L in the grad-u slot) + (the derivative of L in the u slot) = 0. For the Dirichlet energy E[u] = one-half the integral of |grad u|^2, this machinery produces exactly Laplacian u = 0; adding a source term f u produces Poisson's equation.
This is the variational viewpoint that organises a huge swath of elliptic theory. It turns 'solve this PDE' into 'minimise this energy', which is often far easier — you hunt for a minimiser by the direct method and then read off that it solves the equation. It is the engine behind Dirichlet's principle (the harmonic function is the minimiser of the Dirichlet energy), the minimal surface equation (minimise area), and the equations of elasticity and image processing. The deep caveat: a solution of the Euler-Lagrange equation is only a CRITICAL point of the energy — a minimum, a maximum, or a saddle — so finding a critical point is not yet finding a minimiser, and that gap is where much of the subtlety lives.
Minimise the Dirichlet energy E[u] = one-half the integral of (u_x^2 + u_y^2) over a region, among functions with prescribed boundary values. Replace u by u + epsilon v with v = 0 on the boundary; the derivative at epsilon = 0 is the integral of (u_x v_x + u_y v_y). Integrating by parts gives minus the integral of v (u_xx + u_yy). For this to vanish for every v, we need u_xx + u_yy = 0 — the minimiser is harmonic. That is Dirichlet's principle in one line.
Setting the first variation to zero turns 'minimise the energy' into 'solve a PDE'.
A solution of the Euler-Lagrange equation is a critical point, not necessarily a minimiser — it could be a saddle or a maximum. You only know a critical point is a genuine minimum if you check a second-order condition, typically convexity of the Lagrangian in the gradient. Conversely, not every PDE comes from an energy; only those in 'variational' (divergence) form with the right symmetry have an Euler-Lagrange origin.