# Significance of the word 'linear' in linear harmonic oscillator

In my book Advanced Acoustics there is a line-

A particle undergoing SHM is called a linear harmonic oscillator

If I say that the word linear is used for the 2 reasons-

1. The motion of the particle can be defined by a linear differential equation $\frac{d^2x}{dt^2}+\omega ^{2}x=0$
2. Restoring force depends upon the first power of $x$(displacement)

But if we see the case of Damped oscillation the equation of motion is $\frac{d^2x}{dt^2}+2k\frac{dx}{dt}+\omega_{0} ^{2}x=0$ which is also a linear differential equation & the restoring force is dependent on first power of $x$

Now my question is why only Simple Harmonic Oscillator is termed as linear harmonic oscillator & Damped Oscillator not ?

• I'm voting to close this question as off-topic because it is about vocabulary, not physics. – garyp Sep 24 '17 at 20:14
• I think linear means here that the force depends linearly on the position. ($\rightarrow$ the energy depends quadratically). – peterh says reinstate Monica Sep 25 '17 at 2:31

• I believe that's not correct: the damping is not conservative, so how can you associate it with an influence over the potential $V(x)$? – stafusa Sep 25 '17 at 15:06