Euler-Lagrange Equation

Source lecture(s): PHY622 Lec1

Physical Meaning

Stationary condition for a functional integral.

Variable Definitions

Symbol	Meaning
\(F\)	Lagrangian density
\(y\)	dependent variable
\(x\)	independent variable

Assumptions

\(F\) is smooth; endpoints fixed.

Derivation

Perturb \(y\to y+\epsilon\eta\), expand \(J\), set \(\delta J=0\).

Applications

Classical mechanics (Lagrangian formulation), optics (Fermat's principle), general relativity.

Connections to Other Equations

• hamilton-principle
• geodesics