Canonical Transformation
A change of coordinates in phase space (q, p) → (Q, P) that preserves the canonical form of Hamilton's equations.
Governing formula
(q, p) → (Q, P) such that Q̇_i = ∂K / ∂P_i, Ṗ_i = -∂K / ∂Q_i
SI unit
Phase Space Coordinate Transformation
In depth
Canonical transformations preserve phase space volume (Liouville's theorem) and Poisson bracket structures. Hamilton-Jacobi theory uses canonical transformations to transform variables into cyclic constants of motion.
Examples in the real world
Transforming coupled oscillator coordinates into independent uncoupled normal modes.