Worked example - the swap transformation via F1. Take generating function F1(q, Q) = q Q (type F1 depends on old and new coordinates). Rules: p = dF1/dq = Q, and P = -dF1/dQ = -q. So the transformation is Q = p, P = -q: new 'coordinate' is the old momentum and vice versa. Check canonicity: [Q, P] = [p, -q] = -[p, q] = [q, p] = 1. Preserved. The new Hamiltonian K = H + dF1/dt = H (no explicit time dependence), e.g. harmonic oscillator H = p^2/2m + (1/2)k q^2 becomes K = Q^2/2m + (1/2) k P^2 - same form, variables swapped. Hamilton's equations in (Q, P) hold: Q_dot = dK/dP = kP/m... wait, that gives Q_dot = k P/m? Let's verify: dK/dP = kP... (1/2)kP^2 derivative is kP. Hmm, and P_dot = -dK/dQ = -Q/m. Meanwhile from original: q_dot = p/m becomes -P_dot = Q/m, consistent. The equations are preserved - that is canonicity in action.