All small oscillations are harmonic

Harmonic motion is not a special case: it is generic. Every system in stable equilibrium, if displaced slightly and released, oscillates harmonically. The reason lies in the shape of the potential energy near a minimum.

Let us expand the potential energy $E_p(x)$ in a Taylor series about the equilibrium point $x_0$:

$E_p(x) = E_p(x_0) + \underbrace{E_p'(x_0)}_{=\,0}(x-x_0) + \tfrac{1}{2}\,E_p''(x_0)\,(x-x_0)^2 + \cdots$

The linear term is zero because at the equilibrium point the force $F = -E_p'$ is zero (this is the definition of equilibrium). Neglecting the higher-order terms — legitimate for small displacements — the potential energy has the shape of a parabola:

$E_p(x) \approx \tfrac{1}{2}\,k\,(x-x_0)^2, \qquad k = E_p''(x_0)$

and the resulting force is $F = -\dfrac{\dd E_p}{\dd x} = -k(x-x_0)$: identical to Hooke’s law. It is exactly the same harmonic oscillator as the spring.

Principle — Universality of harmonic motion

Near a stable equilibrium point, every system oscillates harmonically with angular frequency $\ev{\omega = \sqrt{\frac{E_p''(x_0)}{m}}}$ The “effective spring constant” $k$ is the second derivative of the potential energy at the equilibrium point: it measures how “steep” the potential well is.

Why it is everywhere

This is why molecules vibrate harmonically about their bonds, and $LC$ circuits “oscillate” like springs: any potential well, seen closely enough, is a parabola. We had already used this idea in the chapter on energy and work, deriving the period of small oscillations directly from the parabola tangent to $U(x)$; here we have given the structural explanation.

Links

Topics: Oscillations and harmonic motion Concepts: Simple harmonic motion · Conservative forces Skills: Small-oscillation approximation Methods: Linear approximation of small oscillations

Related exercises: Worked exercise — Finding A and φ from initial conditions · Problem — Mass on two springs in parallel · Problem — Period and frequency of a mass-spring system