In mathematics, the Dirichlet energy is a measure of how variable a function is.
, subject to appropriate boundary conditions, is equivalent to solving the variational problem of finding a function u that satisfies the boundary conditions and has minimal Dirichlet energy.
In a more general setting, where Ω ⊆ Rn is replaced by any Riemannian manifold M, and u : Ω → R is replaced by u : M → Φ for another (different) Riemannian manifold Φ, the Dirichlet energy is given by the sigma model.
The solutions to the Lagrange equations for the sigma model Lagrangian are those functions u that minimize/maximize the Dirichlet energy.
Restricting this general case back to the specific case of u : Ω → R just shows that the Lagrange equations (or, equivalently, the Hamilton–Jacobi equations) provide the basic tools for obtaining extremal solutions.