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.

Figure $\PageIndex{1}$: The mass on the spring and its equilibrium position

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

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