18: Driven Oscillator
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Michael Fowler (closely following Landau para 22)
Consider a one-dimensional simple harmonic oscillator with a variable external force acting, so the equation of motion is
which would come from the Lagrangian
(Landau “derives” this as the leading order non-constant term in a time-dependent external potential.)
The general solution of the differential equation is
An important case is that of a periodic driving force
But what happens when
\[x(t)=a^{\prime} \cos \left(\omega t+\alpha^{\prime}\right)+\dfrac{f}{m\left(\omega^{2}-\gamma^{2}\right)}[\cos (\gamma t+\beta)-\cos (\omega t+\beta)]\)
The second term now goes to
The amplitude of the oscillations grows linearly with time. Obviously, this small oscillations theory will crash eventually.
But what if the external force frequency is slightly off resonance?
Then (real part understood)
with
The wave amplitude squared
We’re seeing beats, with beat frequency
Energy is exchanged back and forth with the driving external force.