Integral functional with a Lagrangian density

Summary

An integral functional assigns to a function $u$ the integral of an expression $L(x, u, \nabla u)$ built from the point, the value of $u$ and its derivatives. The function $L$ is the Lagrangian density. Most problems of the calculus of variations concern such functionals, for example length, area, energy and action. Their smooth critical points solve the Euler–Lagrange equation.