Consider a numerical integrator $\phi(q, p)$ having $|\text{det } \phi'| = 1$. Can we say the integrator is sympletic, that is that $$ \phi'^\top J^{-1} \phi' = J^{-1} $$ where $$ J^{-1} = \begin{pmatrix} 0 & I \\ -I & 0 \end{pmatrix} $$
In other words, if an integrator has absolute determinant Jacobian equal to $1$ can we say it is sympletic?
