1:35 AM Driven Oscillator  
Driven OscillatorMichael Fowler (closely following Landau para 22) Consider a onedimensional 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 nonconstant term in a timedependent 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) with real. 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 ExchangeWe'll derive a formula for the energy fed into an oscillator by an arbitrary timedependent external force. The equation of motion can be written and defining this is This firstorder 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 timedependent perturbation theory result: are equivalent to the annihilation and creation operators. Damped Driven OscillatorThe 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 nearresonant 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 secondorder 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 .  

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