Calculus of Variations — Mathematics Knowledge System

Overview

The calculus of variations seeks functions that extremise (minimise or maximise) functionals — mappings from spaces of functions to ℝ, typically given by integrals. Its central result, the Euler–Lagrange equation, is a differential equation that must be satisfied by any extremising function. The subject underpins classical mechanics (Hamilton's principle), optics (Fermat's principle), differential geometry (geodesics), and modern optimal control theory.

Intuition

Instead of optimising over numbers or vectors, the calculus of variations optimises over functions. The functional J[y] = ∫ F(x, y, y') dx assigns a number to each curve y(x); we seek the curve for which J is stationary (like finding the flat point of a function, but in an infinite-dimensional space of curves).

Formal Definition

Definition

A functional J: C²([a,b]) → ℝ of the form J[y] = ∫_a^b F(x, y(x), y'(x)) dx with y(a) = yₐ, y(b) = y_b is stationary at y* if for all smooth perturbations η with η(a)=η(b)=0, the first variation δJ[y*;η] = 0.

J[y] = \int_a^b F(x,\, y(x),\, y'(x))\, dx

Basic functional

\delta J[y; \eta] = \frac{d}{d\varepsilon}\bigg|_{\varepsilon=0} J[y + \varepsilon\eta] = \int_a^b \left(\frac{\partial F}{\partial y}\eta + \frac{\partial F}{\partial y'}\eta'\right)dx = 0

First variation

\frac{\partial F}{\partial y} - \frac{d}{dx}\frac{\partial F}{\partial y'} = 0

Euler-Lagrange equation

H = p\dot{q} - L,\quad \dot{q} = \frac{\partial H}{\partial p},\quad \dot{p} = -\frac{\partial H}{\partial q}

Hamilton's equations (Legendre transform of Lagrangian)

Notation

Notation	Meaning
$J[y]$	Functional — maps a function y to a real number
$F(x,y,y')$	Lagrangian density (integrand)
$\delta J$	First variation of J
$\eta$	Test function / perturbation (η(a)=η(b)=0)
$F_y,\; F_{y'}$	Partial derivatives of F with respect to y and y'

Properties

Euler-Lagrange equation

F_y - \frac{d}{dx}F_{y'} = 0 \text{ is a necessary condition for } y^* \text{ to extremise } J[y]

Beltrami identity (autonomous case)

\text{When } F \text{ does not depend explicitly on } x: \; F - y'F_{y'} = \text{const}

Natural boundary conditions

\text{If an endpoint is free, the natural BC is } F_{y'}\big|_{x=a \text{ or } b} = 0

Worked Examples

1
F(y,y') = √(1+y'²) does not depend on y.
$F = \sqrt{1 + (y')^2}$
2
Euler-Lagrange: d/dx(Fy') = 0, so Fy' = y'/√(1+y'²) = c (constant).
$\frac{y'}{\sqrt{1+(y')^2}} = c$
3
This gives y' = constant, so y is linear: y = mx + b.
$y' = c' \implies y = mx + b$
4
Boundary conditions y(0)=0, y(1)=1 give y = x.
$y^*(x) = x$

✓ Answer

The shortest curve is the straight line y = x, as expected geometrically.

Practice Problems

Hardproof writing

Find the catenary — the curve minimising ∫(y√(1+y'²)) dx — using the Beltrami identity.

Common Mistakes

Common Mistake

Assuming the Euler-Lagrange equation gives a minimum (not just a stationary point)

The Euler-Lagrange equation is only a necessary condition. The solution could be a minimum, maximum, or saddle. Sufficiency requires the Legendre-Jacobi conditions.

Quiz

The Euler-Lagrange equation for J[y] = ∫F(x,y,y')dx is:

∂F/∂x = 0∂F/∂y - d/dx(∂F/∂y') = 0∂F/∂y + d/dx(∂F/∂y') = 0d/dx(∂F/∂y) = 0

Historical Background

The brachistochrone problem — find the curve of fastest descent under gravity — was posed by Johann Bernoulli in 1696 and solved by Newton, Leibniz, and the Bernoulli brothers. Euler systematised the subject in 1744 by deriving what we now call the Euler–Lagrange equation. Lagrange reformulated mechanics on this basis in his Mécanique analytique (1788). Weierstrass, Legendre, and Jacobi later supplied rigorous sufficiency conditions.

1696
Bernoulli poses the brachistochrone problem
Johann Bernoulli
1744
Euler derives the Euler–Lagrange equation
Leonhard Euler
1788
Lagrange reformulates mechanics via variational principles
Joseph-Louis Lagrange
1927
Bliss's rigorous treatment in Calculus of Variations
Gilbert Bliss

Summary

The calculus of variations extremises functionals J[y] = ∫F(x,y,y')dx over functions y with given boundary values.
A necessary condition is the Euler-Lagrange equation: Fy - (d/dx)Fy' = 0.
When F does not depend on x, the Beltrami identity F - y'Fy' = const is a first integral.
Classical mechanics, optics (Fermat), geodesics, and optimal control all rest on variational principles.

References

BookGelfand, I.M. and Fomin, S.V. — Calculus of Variations (1963), Dover Publications
WebsiteWikipedia — Calculus of variations
WebsiteMathWorld — Euler-Lagrange Differential Equation