papers/hanzo-hmm/sections/19-convergence-proofs-for-symplectic-integr.tex

\section{Convergence Proofs for Symplectic Integrators}

\label{app:convergence}

\begin{lemma}[Verlet Energy Error]
For the Stormer--Verlet integrator applied to HMM with step size $h$, the energy error satisfies:
\begin{equation}
|\mathcal{H}(\bm{q}_n, \bm{p}_n) - \kappa| \le C h^2 \quad \forall n \le N,
\end{equation}
where $C$ depends on the third derivatives of $\mathcal{H}$ and $N \cdot h = T$ is the total integration time.
\end{lemma}

\begin{proof}
By backward error analysis~\cite{hairer2006geometric}, the Verlet method exactly solves a modified Hamiltonian $\tilde{\mathcal{H}} = \mathcal{H} + h^2\mathcal{H}_2 + O(h^4)$ where $\mathcal{H}_2$ involves third-order terms. Since $\tilde{\mathcal{H}}$ is exactly conserved, $|\mathcal{H}_n - \tilde{\mathcal{H}}_0| = |(\mathcal{H}_n - \tilde{\mathcal{H}}_n) + (\tilde{\mathcal{H}}_n - \tilde{\mathcal{H}}_0)| = |h^2\mathcal{H}_2 + O(h^4)| \le Ch^2$.
\end{proof}

\begin{lemma}[Ruth-4 Energy Error]
For Ruth's 4th-order integrator, the energy error satisfies:
\begin{equation}
|\mathcal{H}(\bm{q}_n, \bm{p}_n) - \kappa| \le C h^4 \quad \forall n \le N.
\end{equation}
\end{lemma}

The proof follows analogously with the modified Hamiltonian $\tilde{\mathcal{H}} = \mathcal{H} + h^4\mathcal{H}_4 + O(h^6)$.