Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 8E / b

Solution

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 $k^{- 1/4}$ .

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