Classical Integrability

Develop a first working command of integrability through Hamiltonian mechanics, Toda, and KdV.

Two starting sequences are available. Open Toda starts with moving particles and develops a finite-dimensional proof of integrability. KdV solitons starts with a travelling wave and develops conservation, auxiliary scattering and reconstruction. Each has local preparation, solved exercises and a reproducible numerical experiment. The KdV two-soliton follow-up then uses a collision to explain elastic scattering and signed position shifts.

For an earlier starting point, use Hamiltonian brackets and canonical coordinates, then solve an oscillator two ways. The harmonic oscillator gives a complete first example of action–angle motion. Continue with Build actions and angles to derive the energy-dependent frequency of a nonlinear quartic oscillator. Both examples exclude their zero-energy equilibrium from the angle coordinates. Apply Liouville–Arnold carefully then tests the theorem’s hypotheses on pendulum librations, rotations and the separatrix. The theorem reference states which conclusions extend to many degrees of freedom.

Choose the system that is more familiar. The comparison is useful later: Toda’s finite Lax matrix and KdV’s differential operator share a spectral idea, but they require different domains, boundary assumptions and completeness arguments. The broader chapter map below identifies both available lessons and planned extensions.