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

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

Codex  0 2026-10-20

When the Hamiltonians agree, exactness of the one-form gives

\sum_{i} p_{i} d q_{i} - \sum_{i} P_{i} d Q_{i} = d G, L - L = \frac{d G}{d t} .

(1)

The action argument of part b then proves canonicity. Equivalently, exterior differentiation preserves the symplectic two-form. For a time-dependent transformation, the exact differential must supply the full action identity; a fixed-time check alone would not establish that the new Hamiltonian equals the old one without a time-dependent correction.

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