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The calculus of variations studies the optimization of functionals—mappings from function spaces to real numbers. Given a functional J[u] = ∫L(x, u, ∇u)dx (the Lagrangian or action), a critical point satisfies the Euler-Lagrange equation: -div(∂L/∂(∇u)) + ∂L/∂u = 0. This transforms the optimization problem into a PDE. Many fundamental PDEs arise this way: Laplace's equation from the Dirichlet energy, minimal surface equations from area functionals, and the equations of elasticity from strain energy. The framework connects PDE theory to physics (principle of least action) and geometry (geodesics, minimal surfaces).
The calculus of variations is the mathematical framework connecting optimization of functionals to PDEs. The central problem is: among all functions u satisfying given boundary conditions, find the one minimizing J[u] = ∫_Ω L(x, u, ∇u)dx. If u is a minimizer and φ is any perturbation vanishing on the boundary, then the function g(ε) = J[u + εφ] has a minimum at ε = 0, so g'(0) = 0. Computing this derivative (the first variation) and using integration by parts yields the Euler-Lagrange equation as the necessary condition for optimality.
For the Dirichlet energy J[u] = ½∫|∇u|²dx, the Euler-Lagrange equation is Laplace's equation Δu = 0. For the area functional J[u] = ∫√(1+|∇u|²)dx, it is the minimal surface equation. For the elastic energy of a beam J[u] = ∫u''²dx, it is the biharmonic equation Δ²u = 0. Each of these PDEs inherits a variational structure: solutions are characterized not just as satisfying a differential equation but as optimizing a geometric or physical quantity.
The direct method of the calculus of variations, perfected by Tonelli, provides existence of minimizers under general conditions. The key hypotheses are: (1) coercivity of J (J[u] → ∞ as ||u|| → ∞, ensuring minimizing sequences are bounded), (2) weak lower semicontinuity of J (the limit of a minimizing sequence achieves a value no larger than the limit of the energies), and (3) convexity of L in ∇u (which ensures weak lower semicontinuity). When L is not convex in ∇u—as occurs in phase transitions, microstructure, and shape memory alloys—minimizers may not exist, and the correct notion is the relaxation of J, replacing L with its convex envelope.
The second variation δ²J determines stability: a critical point is a (local) minimizer if δ²J > 0, which leads to the Jacobi equation and the theory of conjugate points. For multiple critical points of non-convex functionals, topological methods (mountain pass, Ljusternik-Schnirelman, Morse theory) provide existence results. Noether's theorem connects symmetries of the Lagrangian to conservation laws—translational symmetry gives momentum conservation, rotational symmetry gives angular momentum conservation, time symmetry gives energy conservation—providing a deep structural link between the variational formulation and the physics of the problem.