Skip to content

Lagrange Multipliers (Variational)

Source lecture(s): PHY622 Lec1

Intuition

Constraints turn free variation problems into constrained ones by adding a term that penalizes constraint violations.

Formal Definition

For constraint \(G[y]=0\), extremize \(\int (F+\lambda G)\,dx\). The multiplier \(\lambda(x)\) enforces the constraint.

Mathematical Formulation

\[\frac{\partial}{\partial y}(F+\lambda G) - \frac{d}{dx}\frac{\partial}{\partial y'}(F+\lambda G) = 0$$ $$G[y]=0\]

Derivation

Lagrange undetermined multiplier method: require stationarity of \(J+\lambda G\) with respect to both \(y\) and \(\lambda\).

Worked Example

Brachistochrone under gravity with fixed endpoints uses \(F=\sqrt{(1+y'^2)/2gy}\).

Common Mistakes

  • Treating \(\lambda\) as a constant when it can be an arbitrary function of \(x\).
  • Forgetting natural boundary conditions from fixed endpoints.

Quiz

Q1. What does the multiplier \(\lambda\) represent physically?

Answer

A generalized force that enforces the constraint.