Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 2 / 15C / b / Solution

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.