Extremals

Extremals are objects in the calculus of variations that make a given functional stationary under small perturbations. In many problems, one seeks a function or curve that minimizes (or more generally, makes stationary) a quantity expressed as an integral, called a functional. An extremal is a candidate solution to this extremization problem, and it need not be a true minimum or maximum; it may be a saddle point.

A typical setting considers a functional J[y] = ∫_a^b L(x, y(x), y'(x)) dx, with boundary conditions such as y(a) = α and y(b) = β fixed. A function y(x) is an extremal if the first variation δJ vanishes for all admissible variations. This leads to the Euler–Lagrange equation: ∂L/∂y − d/dx (∂L/∂y') = 0. Solutions to these equations are extremals. If endpoints are free or subject to constraints, natural boundary conditions or Lagrange multipliers modify the equations.

Extremals include many familiar objects. For example, the shortest path between two points in Euclidean space is an extremal of the arc-length functional L = √(1 + y'^2), and its solutions are straight lines. In physics and geometry, extremals of various functionals yield geodesics, particle trajectories under conservative forces, or stationary actions in mechanics and field theory.

Higher-order functionals and systems with several dependent variables generalize the Euler–Lagrange framework. Noether’s theorem links symmetries of the functional to conserved quantities along extremals, clarifying the structure of solutions.