Past exam of the theoretical physics I course of the University of Cambridge / 2014 / Paper 1 / 3

For a regular Lagrangian define $p_i=\partial L/\partial {\dot{q}}_i$ and $H={\sum }_i{p}_i{\dot{q}}_i-L$. Its differential isThe velocity differentials cancel. Using $\partial L/\partial q_i={\dot{p}}_i$ and comparing independent coefficients givesThe last equality concerns explicit partial derivatives in their respective natural variables; it does not assert that total time derivatives are always negatives.

The assumed relation makes the old and new actions differ by the endpoint term $G(t_f)-G(t_i)$. Fixed endpoint values of both coordinate sets make its variation zero. Therefore stationarity of the old action is equivalent to stationarity ofVarying independent $Q_i,P_i$ and integrating $P_i\delta {\dot{Q}}_i$ by parts givesBoth coefficients vanish: the new variables satisfy Hamilton's equations, so a locally invertible transformation with this action identity is canonical.

When the Hamiltonians agree, exactness of the one-form givesThe action argument of part b then proves canonicity. Equivalently, exterior differentiation preserves the symplectic two-form. For a time-dependent transformation, the exact differential must supply the full action identity; a fixed-time check alone would not establish that the new Hamiltonian equals the old one without a time-dependent correction.

The general action identity isTreating $q,Q,t$ as independent and comparing coefficients givesChoose a complete integral of the Hamilton-Jacobi equation,Then $\mathcal{H}=0$, and Hamilton's equations become ${\dot{Q}}_i={\dot{P}}_i=0$. Completeness and local invertibility are needed: a solution with too few independent parameters cannot generate an arbitrary trajectory.

Separating $G=W(q)-\beta t$ gives $[W^{\prime }(q){]}^2/(2m)+m{\omega }^2{q}^2/2=\beta$. On the positive-momentum branch chooseWith $Q=\beta$ and $A=\sqrt{2\beta /(m{\omega }^2)}$, differentiation with respect to $Q$ givesSince $P=\gamma$ is constant,The displayed integral is a local positive-$p$ branch. Continuing the trajectory through turning points switches momentum branches and extends the sine solution over the complete orbit. $\beta$ is the oscillator energy, and $\gamma$ fixes its time phase.