Canonical Transformation

A canonical transformation is a change of coordinates in phase space that preserves the fundamental symplectic structure — the Poisson bracket relations — between positions and momenta. In Hamiltonian mechanics, such transformations are the natural generalization of coordinate changes in Lagrangian mechanics, but they are more powerful: they can mix positions and momenta in ways that leave the canonical equations invariant while radically simplifying the Hamiltonian.

The most celebrated canonical transformation is the passage to action-angle variables in integrable systems, which reduces the Hamiltonian to a function of conserved actions alone. This simplification is the gateway to perturbation theory and the KAM theorem, and it reveals that the complexity of a Hamiltonian system is not in its energy function but in the coordinates chosen to describe it.