# Constructing equivalence-preserving Dirac variational integrators with forces

## TL;DR

This paper develops a class of Dirac variational integrators with forces that preserve geometric structures in interconnected mechanical systems, enhancing long-term simulation accuracy.

## Contribution

It introduces a novel construction of equivalence-preserving Dirac variational integrators with forces for interconnected systems, extending variational integrator theory.

## Key findings

- Derived a new class of Dirac variational integrators with force preservation properties
- Demonstrated improved structure preservation in interconnected systems
- Discussed potential applications and future research directions

## Abstract

The dynamical motion of mechanical systems possesses underlying geometric structures, and preserving these structures in numerical integration improves the qualitative accuracy and reduces the long-time error of the simulation. For a single mechanical system, structure preservation can be achieved by adopting the variational integrator construction. This construction has been generalized to more complex systems involving forces or constraints as well as to the setting of Dirac mechanics. Variational integrators have recently been applied to interconnected systems in Parks and Leok (2017), which are an important class of practically useful mechanical systems whose description in terms of Dirac structures and Dirac mechanical systems was elucidated in Jacobs and Yoshimura (2014). Since these interconnected systems are modeled as a collection of subsystems with forces of interconnection, we revisit some of the properties of forced variational integrators. In particular, we derive a class of Dirac variational integrators with forces that exhibit preservation properties that are critical when applying variational integrators to the discretization of interconnected Dirac systems. We close with a discussion of ongoing and future research based on these findings.