 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-4/15b/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

The defining relations for a type-2 generating function give

d (F_{2} - P \cdot Q) = p \cdot d q - P \cdot d Q + \frac{\partial F _{2}}{\partial t} d t .

Hence

p \cdot d q - H d t = P \cdot d Q - (H + \frac{\partial F _{2}}{\partial t}) d t + d (F_{2} - P \cdot Q) .

The exact differential changes the action only by an endpoint term. Thus the transformed variational principle has canonical one-form

P \cdot d Q - K d t

(Q, P)

with

K = H + \partial_{t} F_{2} .

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