 Solution

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

The general action identity is

d G = \sum_{i} p_{i} d q_{i} - \sum_{i} P_{i} d Q_{i} + (H - H) d t .

Treating

q, Q, t

as independent and comparing coefficients gives

p_{i} = G_{q_{i}}, P_{i} = - G_{Q_{i}}, H = H + G_{t} .

G_{t} + H (q, G_{q}, t) = 0.

Then

H = 0

\dot{Q}_{i} = \dot{P}_{i} = 0

. Completeness and local invertibility are needed: a solution with too few independent parameters cannot generate an arbitrary trajectory.

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