Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 7D / a

Solution

The principle of stationary action says that the physical path makes the action stationary under smooth variations

q + η h

with fixed endpoints

h (t_{1}) = h (t_{2}) = 0

. It need not minimize the action. Differentiating and integrating by parts gives

0 = δ S = \int_{t_{1}}^{t_{2}} (L_{q} h + L_{\overset{q}{˙}} \dot{h}) d t = \int_{t_{1}}^{t_{2}} (L_{q} - \frac{d}{d t} L_{\overset{q}{˙}}) h d t .

\frac{d}{d t} \frac{\partial L}{\partial q ˙} - \frac{\partial L}{\partial q} = 0 .

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