Consider a symplectic map $\phi_t(p,q)\in \mathbb{R}^{2n}$, that solves or approximates the Hamiltonian system $$\dot{p} = - \nabla V(q),\quad\dot{q} = M^{-1}p,$$ for Hamiltonian $H = \frac{1}{2}p^TM^{-1}p + V(q)$. That is, the Hamiltonian is separable, quadratic in $p$, $V(q)$ is an arbitrary differentiable function and $M$ is symmetric and constant. What can we say about $\phi_t$ given this information and how can we leverage the structure of this Hamiltonian to find a better approximation to $\phi_h$?
My thoughts so far: For example, given this particular form of the Hamiltonian, we can see that it is an even function of $p$ and therefore has time-reversal symmetry, so it's flow must also be time-reversible (i.e., invariant under the time-reversal operator $R:(p,q,t)\rightarrow(-p,q, -t)$). So the symplectic map must share this reversing symmetry $R\circ\phi_h = \phi_h^{-1}\circ R$.
To leverage the separability property of the Hamiltonian, we can use the Störmer-Verlet map to construct an explicit symplectic mapping (or any symplectic numerical method), given $V(q)$. Such numerical methods can also have time-reversal symmetry, but are only accurate for short times $t<<1$.
To leverage the quadratic in $p$ propery, by defining $\phi_t(p,q)=(P, Q)$, where $P$ and $Q$ can depend on $(p, q, t)$, then using the definition of a symplectic transformation we require $$P_p^TQ_q - P_q^TQ_p = I,$$ where the subscripts denote a partial derivative. As $Q$ must satisfy $\dot{Q}=M^{-1}P$, this becomes $$\dot{Q}_p^TQ_q - \dot{Q}_q^TQ_p = M^{-1}.$$. (preserves the Poisson-bracket) we can then use the chain rule on $\dot{Q}$ to get a complex expression. But I think we need a good ansatz about the form of $Q$ that can solve/approximate the above for $Q$.
I would like to know more maps that can approximate $\phi_h$ by leveraging/preserving the structure of the Hamiltonian, and/or by combining the above ideas.
Any references on this would be appreciated too :)