7.1: Prelude to Harmonic Oscillators

You may be familiar with several examples of harmonic oscillators form classical mechanics, such as particles on a spring or the pendulum for small deviation from equilibrium, etc.

Let me look at the characteristics of one such example, a particle of mass \(m\) on a spring. When the particle moves a distance \(x\) away from the equilibrium position \(x_0\), there will be a restoring force \(-k x\) pushing the particle back ( \(x>0\) right of equilibrium, and \(x<0\) on the left). This can be derived from a potential

\[V(x)=\frac{1}{2} k x^2 .\]

Actually we shall write \(k=m \omega^2\). The equation of motion

\[m \bar{x}=-m \omega^2 x\]

has the solution

\[x(t)=A \cos (\omega t)+B \sin (\omega t) .\]
 

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