 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ii/paper-1/15c/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Consider a smooth variation

q^{a} (t) + ϵ η^{a} (t)

with

η^{a} (t_{1}) = η^{a} (t_{2}) = 0

. Differentiating the action at zero and integrating the velocity term by parts gives

δ S = \int_{t_{1}}^{t_{2}} [\frac{\partial L}{\partial q ^{a}} η^{a} + \frac{\partial L}{\partial q ˙ ^{a}} \overset{η}{˙}^{a}] d t = \int_{t_{1}}^{t_{2}} [\frac{\partial L}{\partial q ^{a}} - \frac{d}{d t} \frac{\partial L}{\partial q ˙ ^{a}}] η^{a} d t .

The boundary term vanishes because the endpoints are fixed. Independent compactly supported variations and the fundamental lemma of the calculus of variations make every bracket zero. Thus the Euler-Lagrange equations are

\frac{d}{d t} \frac{\partial L}{\partial q ˙ ^{a}} = \frac{\partial L}{\partial q ^{a}} .

The required principle is stationarity of the action; an extremal need not be a strict minimum.

 New to topics? Read the docs here!