If $V(x)$ has a stable equilibrium at $x_0$, then expanding gives $V(x)\approx V(x_0)+\tfrac12 k(x-x_0)^2$. The equation of motion becomes $m\ddot x+kx=0$ after shifting coordinates, with angular frequency $\omega_0=\sqrt{k/m}$.

For small displacements $\mathbf{q}$ about equilibrium, the equations often reduce to $M\ddot{\mathbf{q}}+K\mathbf{q}=0$. Seeking solutions $\mathbf{q}=\mathbf{a}e^{i\omega t}$ yields the generalized eigenvalue problem $K\mathbf{a}=\omega^2 M\mathbf{a}$.

A damped driven oscillator satisfies $m\ddot x+b\dot x+kx=F_0\cos\omega t$. The response amplitude exhibits resonance, while the phase shift and quality factor encode dissipation and bandwidth.