Small Oscillations

Expansion Near Equilibrium

Near a stable equilibrium $q_0$, expand the potential:

$$V(q)\simeq V(q_0)+\frac{1}{2}k(q-q_0)^2.$$

Linear Equation

The motion becomes

$$m\ddot x+kx=0,$$

with angular frequency

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

Normal Modes

For many coupled coordinates, diagonalizing the quadratic form reveals independent normal modes.