Euler–Lagrange equation

Summary

The Euler–Lagrange equation of an integral functional $\int L(x, u, \nabla u) \, dx$ is the differential equation satisfied by its smooth critical points, in particular by its smooth minimisers. It is the analogue, for functionals, of the condition that the derivative vanishes at a minimum. Many equations of physics and geometry arise this way. Examples are Laplace's equation from the Dirichlet energy, the minimal surface equation from area, and Newton's equations from the action. It is the basic necessary condition of the calculus of variations.