Skip to content

Euler-Lagrange Equation

Source lecture(s): PHY622 Lec1

Intuition

The condition for a curve to extremize a functional is a differential equation.

Formal Definition

For \(J[y]=\int F(x,y,y')\,dx\), the Euler-Lagrange equation is \(F_y - d/dx(F_{y'}) = 0\).

Mathematical Formulation

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

Derivation

Perturb \(y\to y+\epsilon\eta\), require \(\delta J=0\) for arbitrary \(\eta\), integrate \(\int (F_y\eta + F_{y'}\eta')\,dx\) by parts.

Worked Example

Shortest path in a plane has \(F=\sqrt{1+y'^2}\), giving \(d/dx(y'/\sqrt{1+y'^2})=0\), so \(y'=\) const—a straight line.

Common Mistakes

  • Dropping boundary terms incorrectly.
  • Using \(F\) instead of \(F_y\).

Quiz

Q1. When does the Euler-Lagrange equation reduce to \(F_{y'}=\) const?

Answer

When \(F\) is independent of \(y\).