Lagrange–Dirichlet stability theorem

Remarks

Proof idea: the total energy $E(q, \dot q) = T + V$ is conserved along motions. Since $T$ is positive definite in $\dot q$ and $q_0$ is a strict local minimum of $V$, the function $E - V(q_0)$ is positive near $(q_0, 0)$ except at that point, where it vanishes. So it is a Lyapunov function with zero derivative along motions, and a motion that starts with small energy above $E(q_0, 0)$ stays in a small neighbourhood of $(q_0, 0)$.

Example: the pendulum $L = \tfrac12 m l^2 \dot\theta^2 + m g l \cos\theta$ has $V(\theta) = -m g l \cos\theta$, with a strict minimum at $\theta = 0$, so the hanging position is stable.

The strictness of the minimum cannot be dropped: for a free particle on a line, $V = 0$ has a non-strict minimum at $0$. The motion $q(t) = \varepsilon t$ starts arbitrarily close to $(0, 0)$ but leaves every bounded neighbourhood. The theorem gives stability, not asymptotic stability: without friction, nearby motions oscillate forever and do not approach the equilibrium.

Going deeper

For a student

The theorem is the classical prototype of Lyapunov's direct method, with the total energy as Lyapunov function. It applies to any conservative holonomic system with time-independent constraints, such as pendulums, spring systems and rigid bodies in a potential. Near a nondegenerate minimum, linearisation gives small oscillations with frequencies from the Hessian of $V$ relative to $M(q_0)$. Whether the converse holds, that is, whether an equilibrium that is not a minimum of $V$ must be unstable, is a subtler question with only partial answers.