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Expression of Average energy of Oscillator

What I don't understand is - How they concluded that the average energy should be zero except near resonance - and how that implies that $\omega$ can be replaced by $\omega_{0}$ in this expression. Am I correct in saying that this approximation would only be valid near resonance but the book does not say anything like that. This image is from Kleppner.

enter image description here

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  • $\begingroup$ Your question is not justified. The image text does state that this approximation only applies close to resonance, contrary to your assertion : For $\gamma$ sufficiently small, $\langle E(\omega) \rangle$ is effectively zero except near resonance. $\endgroup$ – sammy gerbil Nov 23 '16 at 0:16
  • $\begingroup$ If you would look at my question - I clearly state that : "but the book does not say anything like that." I simply wrote that line to show my line of thought as required by SE. << How they concluded that the average energy should be zero except near resonance - and how that implies that ω can be replaced by ω0 in this expression. >> - this is what I want to know. $\endgroup$ – Avyakta Purush Nov 23 '16 at 4:24
  • $\begingroup$ I have added the image from the first edition. Note that the graph for heavier damping has been scaled by a factor of 10. It is interesting that the analysis is not done the same way in the second edition? $\endgroup$ – Farcher Nov 23 '16 at 10:24
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The text states that you can see from a sketch that average energy is small except close to resonance. You need to look at a sketch of average energy for the case of light damping.

The text says explicitly that the approximate form of $E(\omega)$ which results is valid only close to $\omega=\omega_0$.

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