Small oscillations about equilibrium

Remarks

The equations are the Lagrange equations of the quadratic Lagrangian $\frac{1}{2} \dot x^T M \dot x - \frac{1}{2} x^T K x$. This is the second-order Taylor approximation of $L - L(q_0, 0)$ at $(q_0, 0)$. Equivalently, they are the linearization of the Lagrange equations of $L$ at the equilibrium. If $K$ is positive definite, $q_0$ is a strict minimum of $V$, every small oscillation is bounded, and it is a sum of normal modes.

Example: the pendulum $L = \frac{1}{2} m \ell^2 \dot \theta^2 + m g \ell \cos \theta$ with $V = -m g \ell \cos \theta$ and $q_0 = 0$. Here $M = m \ell^2$ and $K = m g \ell$, so the equation is $m \ell^2 \ddot x + m g \ell x = 0$, that is $\ddot x + (g/\ell) x = 0$.

Non-example: at the upright position $q_0 = \pi$ the same formula gives $K = -m g \ell < 0$, so the linear equation $\ddot x = (g/\ell) x$ has exponentially growing solutions; they are not oscillations, and the equilibrium is unstable.