Separate simple harmonic oscillator from the damped oscillators

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A simple harmonic oscillator, without damping, can be described by the Lagrangian action
$$S_L[x] = \int_{t_0}^{t_1} \mathbb{d}t \left\{\frac{1}{2} m \dot x^2 - \frac{1}{2} m \omega x^2 \right\}\,.$$
This action is so important that I think it makes sense to single out the simple harmonic oscillator.

I also propose to have damped oscillators as a generalisation (mathematical-physically), or a subclass, of simple harmonic oscillators.

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This separation also enables #10. With damping, I don't know how to start a Hamiltonian description..

[emptymalei]

Makes sense. I can work on this.