 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ii/paper-4/9c/solution

A one-dimensional canonical transformation is a locally invertible change of phase-space variables preserving the symplectic two-form:

d Q \land d P = d q \land d p

. Equivalently it preserves Hamilton equations for the transformed Hamiltonian. Expanding the two-form yields

d Q \land d P = (Q_{q} P_{p} - Q_{p} P_{q}) d q \land d p,

{Q, P} = Q_{q} P_{p} - Q_{p} P_{q} = 1 .

Conversely this Jacobian condition makes the transformation locally invertible and symplectic in two dimensions.

 New to topics? Read the docs here!