Non-Existence of Conjugate-Symplectic Multi-Step Methods of Odd Order

Abstract

We prove that any linear multi-step method $G_1^{\tau}$ of the form $$\sum^{m}_{k=0}\alpha _k Z_{k}=\tau \sum^{m}_{k=0}\beta _k J^{-1}\nabla H(Z_k)$$ with odd order $u$ ($u\ge 3$) cannot be conjugate to a symplectic method $G_2^{\tau}$ of order $w$ ($w\ge u$) via any generalized linear multi-step method $G_3^{\tau}$ of the form $$\sum^m_{k=0} \alpha_k Z_k = \tau\sum^m_{k=0} \beta_k J^{-1}\nabla H(\sum^m_{l=0}\gamma_{kl}Z_l).$$ We also give a necessary condition for this kind of generalized linear multi-step methods to be conjugate-symplectic. We also demonstrate that these results can be easily extended to the case when $G_3^{\tau}$ is a more general operator.