 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-3/8e/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

The adiabatic invariance of the action applies when the frequency changes by only a small fraction during one oscillation and no separatrix is encountered. A convenient condition is

\frac{ω ˙}{ω ^{2}} ≪ 1, ω (t) = \frac{k ( t )}{m} .

Then

I = \frac{E}{ω} ≃ constant .

Consequently

E = I ω \propto k .

A

E = \frac{1}{2} k A^{2}

, and hence

A = \frac{2 I}{mω} \propto ω^{- 1/2} \propto k^{- 1/4} .

Thus a slowly stiffened spring raises the energy as $k^{1/2}$ while reducing the amplitude as $k^{- 1/4}$ .

 New to topics? Read the docs here!