Calculus of variations

A branch of mathematics that finds extrema of functionals, which map functions to numbers, subject to specified constraints.

Functional: A map from a space of functions to numbers. The objects being optimized are functionals rather than ordinary functions of finitely many variables.
Function space: A set of functions equipped with a structure such as a norm or topology. The admissible functions form the domain over which a functional is optimized.
Principle of least action: A physical principle stating that motion makes an action functional stationary. Classical mechanics turns particle trajectories into extrema or stationary points of an action.
Johann Bernoulli: A Swiss mathematician who helped develop early calculus and posed the brachistochrone problem. His 1696 challenge prompted solutions that helped launch systematic variational methods.
Lagrange multiplier: A scalar introduced to incorporate an equality constraint into an optimization objective. Finite-dimensional multipliers have a direct analogue in constrained variational problems.
Euler–Lagrange equation: A differential equation that stationary curves of a broad class of functionals must satisfy. It supplies the central necessary condition for unconstrained extrema.
Admissible function: A function satisfying the regularity, boundary, and other restrictions of a given problem. The choice of admissible functions determines which variations and extrema are meaningful.
Geodesic: A locally length-minimizing or locally straight path in a geometric space. Geodesics arise by extremizing curve length, a basic variational problem.
Leonhard Euler: An eighteenth-century mathematician who made foundational contributions across analysis, mechanics, and geometry. Euler derived a general equation for extrema of integral functionals.
Numerical optimization: The use of computational algorithms to find approximate optima of mathematical objectives. Variational analysis derives conditions on functions, while numerical optimization computes approximations.

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