Past exam of the theoretical physics I course of the University of Cambridge / 2014 / Paper 1 / 3 / b

The assumed relation makes the old and new actions differ by the endpoint term

G (t_{f}) - G (t_{i})

. Fixed endpoint values of both coordinate sets make its variation zero. Therefore stationarity of the old action is equivalent to stationarity of

\int (\sum_{i} P_{i} \dot{Q}_{i} - H (Q, P, t)) d t .

(1)

Varying independent

Q_{i}, P_{i}

and integrating

P_{i} δ \dot{Q}_{i}

by parts gives

δ S = \int \sum_{i} [(\dot{Q}_{i} - H_{P_{i}}) δ P_{i} - (\dot{P}_{i} + H_{Q_{i}}) δ Q_{i}] d t .

(2)

Both coefficients vanish: the new variables satisfy Hamilton's equations, so a locally invertible transformation with this action identity is canonical.

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Past exam of the theoretical physics I course of the University of Cambridge / 2014 / Paper 1 / 3 / b

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