Discrete dynamics versus analytic dynamics

Abstract

For discrete classical Molecular dynamics (MD) obtained by the "Verlet" algorithm (VA) with the time increment $h$ there exists a shadow Hamiltonian $\tilde{H}$ with energy $\tilde{E}(h)$, for which the discrete particle positions lie on the analytic trajectories for $\tilde{H}$. Here we proof that there, independent of such an analytic analogy, exists an exact hidden energy invariance $E^*$ for VA dynamics. The fact that the discrete VA dynamics has the same invariances as Newtonian dynamics raises the question, which of the formulations that are correct, or alternatively, the most appropriate formulation of classical dynamics. In this context the relation between the discrete VA dynamics and the (general) discrete dynamics investigated by T. D. Lee [Phys. Lett. $\textbf{122B}$, 217 (1983)] is presented and discussed.