Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 2 / 15C (Classical Dynamics)

Hamilton's equations generate the phase-space vector field $(H_{p_1},\dots ,H_{p_n},-H_{q_1},\dots ,-H_{q_n})$. Its divergence vanishes:by equality of mixed derivatives. The Jacobian $J(t)$ of the flow obeys $\dot{J}=(\mathrm{div}V)J=0$ with $J(0)=1$. Consequently $d\tau =d{q}_1\cdots d{q}_{n}d{p}_1\cdots d{p}_n$ is preserved for every time on which the smooth flow exists. This proves Liouville theorem in Hamiltonian mechanics directly; explicit time dependence of the Hamiltonian would not change the divergence argument.

For a time-independent function $F$, Hamilton's equations giveThus a constant of motion has $[F,H]=0$. For $F={\sum }_k{p}_k$ and the given Hamiltonian, $[F,H]=-{\sum }_k{V}_{q_k}$. Simultaneous translation $q_k\mapsto q_k+s$ leaves every difference $q_i-q_j$, and hence $V$, unchanged. Differentiation with respect to $s$ gives ${\sum }_k{V}_{q_k}=0$, so $[F,H]=0$ also in this example. This is conservation of total momentum from translational symmetry.

A canonical transformation preserves the symplectic form and therefore the form of Hamilton's equations for arbitrary Hamiltonians. In one degree of freedom,Thus the transformation is canonical exactly when $[Q,P]=1$. Equivalently the chain rule gives $[F,G{]}_{q,p}=[Q,P{]}_{q,p}[F,G{]}_{Q,P}$ for arbitrary transformed functions, so the bracket and Hamiltonian dynamics are preserved precisely with this factor one. The condition also makes the Jacobian nonzero and hence supplies local invertibility; a global canonical change of variables additionally requires the stipulated global bijectivity on its domain.