Harmonic approximation ($V(x)\simeq V(x_0)+\frac{1}{2}{V}^{\prime \prime }(x_0)(x-x_0{)}^2$)

Near a stable equilibrium $x_0$ with $V^{\prime }(x_0)=0$ and $V^{\prime \prime }(x_0)>0$, the harmonic approximation truncates the Taylor series of the potential energy at quadratic order. With constant mass $m$, Newton's second law then gives a harmonic oscillator with angular frequency $\omega =\sqrt{V^{\prime \prime }(x_0)/m}$. The approximation requires displacements small enough that the omitted terms are negligible; it does not describe a flat minimum where $V^{\prime \prime }(x_0)=0$.