(127)

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

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

(128)

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

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

(129)
##### Derivation of the Equation of Motion by ...

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