Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 140 / 3 / b

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- Solution b

Solution

Use the Hamiltonian vector field convention

ι_{X_{t}} ω = d H_{t}

, which is forced by the sign of the formula in this part. This is the opposite of the default sign in the linked general article. Since

d λ = - ω

, Cartan's magic formula gives

L_{X_{t}} λ = d (ι_{X_{t}} λ) + ι_{X_{t}} d λ = d (ι_{X_{t}} λ - H_{t}) .

Differentiate the pullback of a differential form along the smooth isotopy and integrate from

0

to

t

:

ρ_{t}^{*} λ - λ = \int_{0}^{t} ρ_{s}^{*} L_{X_{s}} λ d s = d \int_{0}^{t} (ι_{X_{s}} λ - H_{s}) \circ ρ_{s} d s .

Thus the required exact differential form has the explicit primitive

F_{t} = \int_{0}^{t} (ι_{X_{s}} λ - H_{s}) \circ ρ_{s} d s, ρ_{t}^{*} λ - λ = d F_{t} .

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