Calculus of Variations

Source lecture(s): PHY622 Lec1-2

Intuition

Instead of finding maxima/minima of a function, we find curves that make a functional stationary.

Formal Definition

Given a functional \(J[y]=\int_{x_0}^{x_1} F(x,y,y')\,dx\), the extremal satisfies the Euler-Lagrange equation.

Mathematical Formulation

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

Derivation

Consider a variation \(y(x)\to y(x)+\epsilon\eta(x)\) with fixed endpoints. First-order change \(\delta J=0\) for arbitrary \(\eta\) yields the EL equation after integration by parts.

Worked Example

For \(F=y'^2\), EL gives \(y''=0\), so the extremal is a straight line.

Common Mistakes

Forgetting the \(d/dx\) acting on \(\partial F/\partial y'\).
Applying EL when \(F\) depends on higher derivatives (\(y''\)) without modification.

Quiz

Q1. What does 'stationary' mean in calculus of variations?

Answer

\(\delta J=0\); it could be minimum, maximum, or saddle.