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 , where , the solution of the homogeneous equation, and is some particular integral of the inhomogeneous equation.
An important case is that of a periodic driving force A trial solution yields so
But what happens when ? To find out, take part of the first solution into the second, that is,
The second term now goes to as , so becomes the ratio of its first derivatives with respect to (or, equivalently, ).
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)
The wave amplitude squared
We're seeing beats, with beat frequency Note that if the oscillator begins at the origin, then and the amplitude periodically goes to zero, this evidently only occurs when
Energy is exchanged back and forth with the driving external force.
More General Energy Exchange
We'll derive a formula for the energy fed into an oscillator by an arbitrary time-dependent external force.
The equation of motion can be written
and defining this is
This first-order equation integrates to
The energy of the oscillator is
So if we drive the oscillator over all time, with beginning energy zero,
This is equivalent to the quantum mechanical time-dependent perturbation theory result: are equivalent to the annihilation and creation operators.
Damped Driven Oscillator
The linear damped driven oscillator:
(Following Landau's notation here -- note it means the actual frictional drag force is )
Looking near resonance for steady state solutions at the driving frequency, with amplitude , phase lag , that is, , we find
For a near-resonant driving frequency , and assuming the damping to be sufficiently small that we can drop the term along with , the leading order terms give
so the response, the dependence of amplitude of oscillation on frequency, is to this accuracy
(We might also note that the resonant frequency is itself lowered by the damping, but this is another second-order effect we ignore here.)
The rate of absorption of energy equals the frictional loss. The friction force on the mass moving at is doing work at a rate:
The half width of the resonance curve as a function of driving frequency is given by the damping. The total area under the curve is independent of damping.
For future use, we'll write the above equation for the amplitude as