Is Symplectic-Energy-Momentum Integration Well-Posed?

TL;DR

This paper investigates the well-posedness of symplectic-energy-momentum integrators by analyzing the existence and uniqueness of solutions to the discrete-time Hamilton equations, highlighting conditions where solutions may not exist.

Contribution

It provides new theoretical results on the existence and uniqueness of solutions for SEM integrators, identifying conditions where solutions are not guaranteed.

Findings

Certain points in extended-phase space lack solutions for small time steps

The nonlinear pendulum illustrates the main theoretical ideas

Conditions for well-posedness of SEM integrators are characterized

Abstract

We provide new existence and uniqueness results for the discrete-time Hamilton (DTH) equations of a symplectic-energy-momentum (SEM) integrator. In particular, we identify points in extended-phase space where the DTH equations of SEM integration have no solution for arbitrarily small time steps. We use the nonlinear pendulum to illustrate the main ideas.