A generating function of the first type is F_1(q,Q) = qQ^2.
PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Hamiltonian mechanics and canonical structure · Hamiltonian mechanics and canonical structure
Action and Symmetry, Ch Canonical transformations, generating functions and symplectic structure
Problem
A generating function of the first type is F1(q,Q)=qQ2.
((a)) Using p=∂F1/∂q and P=−∂F1/∂Q, and then eliminating Q, obtain Q as a function of q and p on the branch p>0, Q>0, and say what goes wrong at p=0.
((b)) Obtain P as a function of q and p.
((c)) Compute {Q,P} directly in (q,p) and confirm the transformation is canonical.