A Fundamental Theorem on the Structure of Symplectic Integrators

TL;DR

This paper presents a fundamental theorem that characterizes the structure of symplectic integrators, showing limitations on positive-coefficient methods and providing a way to analytically derive higher-order algorithms.

Contribution

It introduces a precise theorem governing symplectic integrator structure, establishing bounds and enabling analytical derivation of fourth-order algorithms.

Findings

Positive-coefficient symplectic integrators cannot exceed second order.

Sharp bounds on second-order error coefficients are derived.

Fourth-order algorithms can be analytically constructed by saturating bounds.

Abstract

I show that the basic structure of symplectic integrators is governed by a theorem which states {\it precisely}, how symplectic integrators with positive coefficients cannot be corrected beyond second order. All previous known results can now be derived quantitatively from this theorem. The theorem provided sharp bounds on second-order error coefficients explicitly in terms of factorization coefficients. By saturating these bounds, one can derive fourth-order algorithms analytically with arbitrary numbers of operators.