The Euler-Lagrange Equation is the major formula of the Calculus of variations. It provides a way to solve for functions which extremize a given cost functional. It is widely used to solve optimization problems, and in conjunction with the action principle to calculate trajectories. It is analogous to the result from Calculus that a function attains its extreme values when its derivative vanishes.
must also satisfy the ordinary differential equation
A standard example is finding the shortest path between two points in the plane. Assume that the points to be connected are (a,c) and (b,d). The length of a path y=f(x) between these two points is:
The Euler-Lagrange equation yields the differential equation:
In other words, a straight line.
There are also various multi-dimensional versions of the Euler-Lagrange equations. If q is a path in n-dimensional space, then it extremizes the cost functional
only if it satisfies
This formulation is particularly useful in physics when L is taken to be be the Lagrangian.
Another multi-dimensional generalization comes from considering a function on n variables. If Ω is some surface, then
is extremized only if f satisfies the partial differential equation
When n=2 and L is the energy functional, this leads to the soap-film minimal surface problem.