The Midpoint Rule as a Variational--Symplectic Integrator. I. Hamiltonian Systems

TL;DR

This paper analyzes the midpoint rule as a variational--symplectic integrator for Hamiltonian systems, demonstrating its preservation of symplectic structure, conservation of Noether charges, and excellent long-term energy behavior, which is promising for applications in general relativity.

Contribution

It establishes that the midpoint rule preserves symplectic form and conserves a near-Hamiltonian phase space function, providing a foundation for applying variational and symplectic integrators to constrained Hamiltonian systems.

Findings

Midpoint rule preserves symplectic form.

It conserves Noether charges.

Exhibits excellent long-term energy behavior.

Abstract

Numerical algorithms based on variational and symplectic integrators exhibit special features that make them promising candidates for application to general relativity and other constrained Hamiltonian systems. This paper lays part of the foundation for such applications. The midpoint rule for Hamilton's equations is examined from the perspectives of variational and symplectic integrators. It is shown that the midpoint rule preserves the symplectic form, conserves Noether charges, and exhibits excellent long--term energy behavior. The energy behavior is explained by the result, shown here, that the midpoint rule exactly conserves a phase space function that is close to the Hamiltonian. The presentation includes several examples.