Classical Hamiltonian Systems

Develop finite-dimensional integrable dynamics, its constructions, global geometry and obstructions.

Begin with the finite open Toda derivation: it distinguishes isospectral invariants from the involution and independence needed for Liouville integrability. The Toda learning sequence supplies a slower route with worked examples and practice, and the model record fixes its boundary regime.

The Liouville–Arnold result distinguishes this integrability test from the additional hypotheses needed for invariant tori. For explicit action–angle calculations, work through the harmonic oscillator and then the nonlinear quartic oscillator. The second calculation derives its action, period and energy-dependent frequency and handles the angle’s turning-point branches. The pendulum lesson then tests regularity, connectedness and compactness across different energy regimes and configuration spaces.

The chapter map below places this example among the broader constructions, global questions and obstructions planned for the volume. Linked readings are available; plain-text titles are future coverage.