From modelling of systems with constraints to generalized geometry and back to numerics

TL;DR

This paper explores how generalized geometry objects like Dirac structures can be used to develop new numerical methods, called Dirac integrators, for simulating constrained mechanical systems more accurately.

Contribution

It introduces Dirac integrators based on generalized geometry, demonstrating their advantages in preserving constraints and influencing qualitative analysis of mechanical systems.

Findings

Dirac integrators better preserve constraints in simulations

Choice of numerical method affects qualitative dynamics conclusions

Dirac integrators relate to geometric stability concepts

Abstract

In this note we describe how some objects from generalized geometry appear in the qualitative analysis and numerical simulation of mechanical systems. In particular we discuss double vector bundles and Dirac structures. It turns out that those objects can be naturally associated to systems with constraints -- we recall the mathematical construction in the context of so called implicit Lagrangian systems. We explain how they can be used to produce new numerical methods, that we call Dirac integrators. On a test example of a simple pendulum in a gravity field we compare the Dirac integrators with classical explicit and implicit methods, we pay special attention to conservation of constrains. Then, on a more advanced example of the Ziegler column we show that the choice of numerical methods can indeed affect the conclusions of qualitative analysis of the dynamics of mechanical systems. We also tell why we think that Dirac integrators are appropriate for this kind of systems by explaining the relation with the notions of geometric degree of non-conservativity and kinematic structural stability.