Near a stable equilibrium, one expands the potential as $V(q)\approx V(q_0)+\tfrac12 \sum_{ij} K_{ij}\eta_i\eta_j$. Diagonalizing the Hessian yields normal modes and frequencies. The linearized problem is often the first step in understanding nonlinear dynamics, metastability, and resonance phenomena.

Nonlinear Newtonian systems can exhibit sensitive dependence on initial conditions, positive Lyapunov exponents, and intricate phase-space structures. Even simple models such as the driven pendulum or restricted three-body problem can produce chaotic motion, making prediction probabilistic or statistical rather than exact at long times.

When a system is close to an integrable one, perturbation theory, averaging, and canonical transformations can reveal slow drifts and resonance overlap. KAM theory explains the persistence of many invariant tori under small perturbations, while resonant islands and separatrices organize the breakdown of regular motion.