---
tag: PQ25NHV0KS
kind: concept
name: Calculus of variations
msc: [49-XX, 49K10, 35A15]
sources:
  - url: https://en.wikipedia.org/wiki/Calculus_of_variations
same_as:
  wikidata: Q216861
  wikipedia: Calculus of variations
---

## Summary

The calculus of variations studies the extreme values of functionals, that is, of maps that send functions to numbers.
Examples of functionals are length, area and energy.
A typical functional is an integral $\int_\Omega L(x, u, \nabla u) \, dx$, minimised among functions $u$ with given boundary values.
Its smooth critical points solve a partial differential equation, the [[Euler–Lagrange equation|PS0VV149ET]].
Conversely, many equations of physics and geometry are Euler–Lagrange equations, and the calculus of variations gives existence through the [[direct method|TE379HBTY0]].

## Explanations

### L1 human

Ordinary calculus finds the point where a function of a number is smallest.
The calculus of variations finds the curve, surface or function that makes a quantity smallest.
Examples are the shortest path between two points, the soap film of least area spanning a wire, and the shape of a hanging chain of least potential energy.
For the shortest path in the plane, the functional is the length $\int_a^b \sqrt{1 + y'(x)^2} \, dx$, and the answer is the straight line.