Return to Question

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 $$ \begin{align} Q &= q + {\Delta}t\frac{\partial H}{\partial p}(q,p),\\ P &= p - {\Delta}t\frac{\partial H}{\partial q}(q,p) \end{align} $$ are not symplectic, while $$ \begin{align} Q &= q - {\Delta}t\frac{\partial H}{\partial p}(q,p),\\ P &= p + {\Delta}t\frac{\partial H}{\partial q}(Q,p) \end{align} $$ are symplectic?

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 $$ \begin{align} Q &= q + {\Delta}t\frac{\partial H}{\partial p}(q,p),\\ P &= p - {\Delta}t\frac{\partial H}{\partial q}(q,p) \end{align} $$ are not symplectic, while $$ \begin{align} Q &= q - {\Delta}t\frac{\partial H}{\partial p}(q,p),\\ P &= p + {\Delta}t\frac{\partial H}{\partial Q}(Q,p) \end{align} $$ are symplectic?

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 $$ \begin{align} Q &= q + {\Delta}t\frac{\partial H}{\partial p}(q,p),\\ P &= p - {\Delta}t\frac{\partial H}{\partial q}(q,p) \end{align} $$ are not symplectic, while $$ \begin{align} Q &= q - {\Delta}t\frac{\partial H}{\partial p}(q,p),\\ P &= p + {\Delta}t\frac{\partial H}{\partial q}(q,p) \end{align} $$ are symplectic?

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 $$ \begin{align} Q &= q + {\Delta}t\frac{\partial H}{\partial p}(q,p),\\ P &= p - {\Delta}t\frac{\partial H}{\partial q}(q,p) \end{align} $$ are not symplectic, while $$ \begin{align} Q &= q - {\Delta}t\frac{\partial H}{\partial p}(q,p),\\ P &= p + {\Delta}t\frac{\partial H}{\partial q}(Q,p) \end{align} $$ are symplectic?