Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 32C / e

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- Solution e

Solution

No. Consider two uncoupled oscillators with nonzero amplitudes:

q_{i} (t) = A_{i} cos (ω_{i} t + ϕ_{i}), p_{i} (t) = - A_{i} ω_{i} sin (ω_{i} t + ϕ_{i}), i = 1, 2.

The orbit is bounded. If it had a period

T > 0

, then

ω_{1} T

and

ω_{2} T

would both be integer multiples of

2 π

ω_{1} / ω_{2} \in Q

. Choosing an irrational frequency ratio gives an incommensurate harmonic-oscillator orbit, which is quasiperiodic on an invariant torus and not periodic.

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