Near a nondegenerate stable equilibrium, many nonlinear mechanical systems behave approximately like harmonic oscillators.

For one-dimensional conservative motion, let $x_0$ be a stable equilibrium and define the small displacement

$$q=x-x_0.$$

At the equilibrium,

$$U'(x_0)=0,$$

and for a nondegenerate minimum,

$$U''(x_0)>0.$$

Expand the potential energy about $x_0$:

$$U(x_0+q) =U(x_0) +U'(x_0)q +\frac12U''(x_0)q^2 +\frac1{3!}U'''(x_0)q^3+\cdots.$$

The linear term vanishes because $x_0$ is an equilibrium. For sufficiently small $|q|$, the leading change in potential is therefore

$$U(x_0+q)\approx U(x_0)+\frac12k_{\rm eff}q^2,$$

where

$$\boxed{k_{\rm eff}=U''(x_0)}.$$

Using $F=-dU/dx$, the corresponding force is

$$F\approx-k_{\rm eff}q.$$

Newton's second law then gives

$$m\ddot q=-k_{\rm eff}q,$$

or

$$\ddot q+\frac{k_{\rm eff}}{m}q=0.$$

Thus the local motion is approximately harmonic with

$$\boxed{\omega=\sqrt{\frac{k_{\rm eff}}{m}}}.$$

The small-oscillation frequency is determined by the curvature of the potential well and the inertia of the moving object.

Worked example: a nonlinear potential

Suppose

$$U(x)=\frac12kx^2+\lambda x^4,$$

with $k>0$ and $\lambda>0$. The stable equilibrium is at $x_0=0$, and

$$U''(0)=k.$$

Therefore the small-amplitude motion has

$$\omega\approx\sqrt{\frac{k}{m}}.$$

The quartic term does not affect the leading small-amplitude frequency because it contributes only at higher order near the equilibrium. At larger amplitudes, however, it becomes important, so the motion is no longer exactly sinusoidal and its period can depend on amplitude.

When the harmonic approximation fails

A stable equilibrium does not always have nonzero quadratic curvature. For example,

$$U(x)=\lambda x^4$$

has a stable minimum at $x=0$ but

$$U''(0)=0.$$

There is then no quadratic restoring term, so the leading motion is not described by a harmonic oscillator. Likewise, even when $U''(x_0)>0$, sufficiently large excursions make the higher-order terms in the Taylor expansion important.

The harmonic approximation is therefore a local approximation near a nondegenerate stable equilibrium, not a claim that the full system is linear. The same idea extends to angular and multi-coordinate systems, where linearization near equilibrium leads to the normal small-oscillation problem.