The calculus of variations extends ordinary optimisation from finding numbers or points that minimise a function, to finding entire functions or curves that minimise (or make stationary) a quantity called a functional β an integral depending on the unknown function itself. Classic problems include the brachistochrone (fastest sliding path under gravity), minimal surfaces, and geodesics (shortest paths on curved surfaces). The field underlies Lagrangian and Hamiltonian mechanics, optimal control theory, and the action principles at the heart of modern physics.
Ordinary calculus finds the minimum of a function: what value of x makes f(x) smallest? The calculus of variations asks a harder question: what entire curve makes some quantity smallest? "Of all the paths between two points on a curved surface, which is the shortest?" "Of all the curves connecting two given points, which gives the fastest slide under gravity?" These are not questions about a single number β they are questions about choosing the right function from an infinite-dimensional space of candidates.
Fermat's principle states that light travelling between two points takes the path that minimises travel time β which is why light bends (refracts) when passing between media of different densities. Snell's law of refraction is a direct consequence of applying variational reasoning to this principle. The calculus of variations was, in this sense, already implicit in 17th-century optics, long before it was fully formalised as a mathematical discipline.
A functional assigns a number to each function in some class β for example, J[y] = β«F(x, y, y') dx assigns to each curve y(x) the value of this integral. The calculus of variations seeks the function y that makes J[y] stationary (typically a minimum or maximum). The analogy with ordinary calculus is exact: just as df/dx = 0 is the condition for a stationary point of a function, the variational derivative Ξ΄J/Ξ΄y = 0 is the condition for a stationary function of a functional.
The fundamental result of the calculus of variations is the Euler-Lagrange equation (derived by Euler in 1744 and Lagrange in 1755): for J[y] = β«F(x, y, y') dx, the stationary function satisfies:
This is a differential equation for the unknown function y β solving it gives the optimising curve. The Lagrange equations of mechanics (see the Hamiltonian & Lagrangian Mechanics entry) are exactly this equation applied to L = T β V.
The shortest path between two points on a plane is a straight line β confirmed immediately by the Euler-Lagrange equation applied to arc length. On a sphere, shortest paths are great circles (geodesics). The minimal surface spanning a given boundary curve (the Plateau problem β see below) satisfies a nonlinear partial differential equation. The isoperimetric problem β the curve of fixed length enclosing maximum area β is a circle, proved rigorously using variational methods. Each of these classic results, known geometrically for millennia, receives its most transparent proof via the calculus of variations.
Plateau's problem asks: given a closed curve in space, does there exist a surface of minimal area spanning it? The problem is named after Joseph Plateau, who observed experimentally in the 1840s (using soap films β which physically minimise surface area subject to boundary constraints) that minimal surfaces always exist for any reasonably shaped boundary curve. The mathematical proof of existence for arbitrary rectifiable curves was given simultaneously, and independently, by Jesse Douglas and Tibor RadΓ³ in 1930β1931 β a result for which Douglas received one of the inaugural Fields Medals in 1936. Minimal surfaces remain a rich and active area of differential geometry and geometric analysis to this day.
Just as in ordinary calculus, finding a stationary point via the Euler-Lagrange equation doesn't guarantee a minimum β it could be a maximum or saddle. The second variation of a functional (the analogue of the second derivative) determines stability. The Jacobi condition and the concept of conjugate points characterise whether a stationary solution is a genuine local minimum of the functional β and their violation indicates the onset of instability, with deep connections to catastrophe theory and bifurcation phenomena.
Optimal control theory, developed substantially by Pontryagin and his collaborators in the USSR in the 1950sβ1960s, extends the calculus of variations to problems where the "curve" to be optimised is the trajectory of a dynamical system under the influence of a controllable input. Pontryagin's Maximum Principle β the central result β gives necessary conditions for an optimal control strategy, generalising the Euler-Lagrange equation to systems with control constraints. Optimal control theory has extensive practical applications in aerospace engineering, economics (optimal resource allocation), robotics, and the design of optimal medical treatment protocols, making the calculus of variations one of the most practically consequential branches of pure mathematics.