Substituting $u(t)=\stackrel{\u02c6}{u}\mathrm{cos}\omega t$, we write

$E=T(t)+V(t)=\frac{1}{2}m{\omega}^{2}{\stackrel{\u02c6}{u}}^{2}{\mathrm{sin}}^{2}\omega t+\frac{1}{2}k{\stackrel{\u02c6}{u}}^{2}{\mathrm{cos}}^{2}\omega t$ (128)

Recalling Eq. (15), i.e., $k={\omega}_{n}^{2}m$, we can write Eq. (128) as

$E=T(t)+V(t)=\frac{1}{2}m{\stackrel{\u02c6}{u}}^{2}({\omega}^{2}{\mathrm{sin}}^{2}\omega t+{\omega}_{n}^{2}{\mathrm{cos}}^{2}\omega t)$ (129)

##### Derivation of the Equation of Motion by ...

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