We know that $dot{q} = frac{partial H}{partial p}$ and $dot{p} = -frac{partial H}{partial q}$, and we also know the values $Q$ and $P$ respectively of $q$ and $p$ at a later time step $Delta t$. How could we prove that the quantities

$$

begin{align}

Q &= q + {Delta}tfrac{partial H}{partial p}(q,p),\

P &= p – {Delta}tfrac{partial H}{partial q}(q,p)

end{align}

$$

are not symplectic, while

$$

begin{align}

Q &= q – {Delta}tfrac{partial H}{partial p}(q,p),\

P &= p + {Delta}tfrac{partial H}{partial Q}(Q,p)

end{align}

$$

are symplectic?

Clarification:

The sets of equations define different numerical integrators: in the first case (qi+1,pi+1) directly in terms of (qi,pi), and in the second case qi+1 in terms of (qi,pi), and pi+1 in terms of (qi+1,pi).