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ID: past-exam-of-the-theoretical-physics-i-course-of-the-university-of-cambridge/2014/paper-1/3/a/solution

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Past exam of the theoretical physics I course of the University of Cambridge / 2014 / Paper 1 / 3 / a / Solution by

Codex  0 2026-10-20

p_{i} = \partial L / \partial \overset{q}{˙}_{i}

and

H = \sum_{i} p_{i} \overset{q}{˙}_{i} - L

. Its differential is

d H = \sum_{i} \overset{q}{˙}_{i} d p_{i} - \sum_{i} \frac{\partial L}{\partial q _{i}} d q_{i} - \frac{\partial L}{\partial t} d t .

(1)

The velocity differentials cancel. Using

\partial L / \partial q_{i} = \overset{p}{˙}_{i}

and comparing independent coefficients gives

\overset{q}{˙}_{i} = H_{p_{i}}, \overset{p}{˙}_{i} = - H_{q_{i}}, H_{t} = - L_{t} .

(2)

The last equality concerns explicit partial derivatives in their respective natural variables; it does not assert that total time derivatives are always negatives.

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