the nonlinear pendulum

Everyone meets the pendulum first in the convenient lie that it swings in a perfect sine wave. That is only true for tiny swings. The honest pendulum — released from a wide angle, or whirled all the way over the top — does something richer, and capturing that richness requires keeping the full nonlinear equation instead of its small-angle simplification. The nonlinear pendulum is the canonical first example of nonlinear dynamics.

Newton's law for a bob on a rigid rod gives theta'' + (g/L) sin(theta) = 0, where theta is the angle from straight down. The nonlinearity is the sin(theta): for small angles sin(theta) is approximately theta, which is the linear simple-harmonic model with its constant period, but for large angles the restoring force grows more slowly than the angle, so the swing takes longer. Written as a planar system with omega = theta', it becomes theta' = omega, omega' = -(g/L) sin(theta). Without friction it is Hamiltonian — total energy is conserved — so the phase portrait is just the contour map of the energy: closed loops around the downward equilibrium (back-and-forth oscillation) and wavy open curves above a threshold (the bob going over the top, round and round).

Through the lens of this field, the pendulum is a gallery of the key ideas. The downward rest (theta = 0) is a center of the conservative system — Lyapunov stable but not asymptotically stable, since energy never drains. The inverted balance point (theta = pi) is a saddle — unstable, exactly the pencil-on-its-tip. The dividing trajectory between swinging and spinning is the separatrix through the saddle. Add damping (a -b omega term) and energy now decays: the center becomes an asymptotically stable spiral, and LaSalle's principle proves the bob must finally come to rest hanging straight down. Honest caveat: the small-angle 'constant period' is only an approximation; a real large-amplitude swing has a longer, amplitude-dependent period.