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Discrete Dirac structures and discrete Lagrange--Dirac dynamical systems in mechanics

Linyu Peng, Hiroaki Yoshimura

TL;DR

This paper develops a comprehensive discrete Dirac framework by introducing $(\pm)$-discrete Dirac structures on manifolds and their induced structures on $T^*Q$, enabling a discrete Lagrange--Dirac formalism. It defines discrete two-forms, discrete constraint spaces, and $(\pm)$-finite difference maps, then constructs Tulczyjew-like discrete triples and Dirac differentials to derive $(\pm)$-discrete Lagrange--Dirac equations via $(\pm)$-discrete Lagrange--d'Alembert--Pontryagin principles. The authors show equivalence of the resulting equations with established discrete Lagrange--d'Alembert equations, ensure preservation of the $(\pm)$-discrete induced Dirac structures in nonholonomic integrators, and validate the approach with numerical tests on a vertical rolling disk and a classical Heisenberg system. The work provides a robust, structure-preserving discretization strategy for nonholonomic mechanics and lays groundwork for extensions to discrete Hamilton–Dirac systems and higher-dimensional/thermodynamic settings. Overall, it offers a unified geometric and variational pathway to accurately simulate constrained mechanical systems in a discrete setting.

Abstract

In this paper, we propose the concept of $(\pm)$-discrete Dirac structures over a manifold, where we define $(\pm)$-discrete two-forms on the manifold and incorporate discrete constraints using $(\pm)$-finite difference maps. Specifically, we develop $(\pm)$-discrete induced Dirac structures as discrete analogues of the induced Dirac structure on the cotangent bundle over a configuration manifold, as described by Yoshimura and Marsden (2006). We demonstrate that $(\pm)$-discrete Lagrange--Dirac systems can be naturally formulated in conjunction with the $(\pm)$-induced Dirac structure on the cotangent bundle. Furthermore, we show that the resulting equations of motion are equivalent to the $(\pm)$-discrete Lagrange--d'Alembert equations proposed in Cortés and Martínez (2001) and McLachlan and Perlmutter (2006). We also clarify the variational structures of the discrete Lagrange--Dirac dynamical systems within the framework of the $(\pm)$-discrete Lagrange--d'Alembert--Pontryagin principle. Finally, we validate the proposed discrete Lagrange--Dirac systems with some illustrative examples of nonholonomic systems through numerical tests.

Discrete Dirac structures and discrete Lagrange--Dirac dynamical systems in mechanics

TL;DR

This paper develops a comprehensive discrete Dirac framework by introducing -discrete Dirac structures on manifolds and their induced structures on , enabling a discrete Lagrange--Dirac formalism. It defines discrete two-forms, discrete constraint spaces, and -finite difference maps, then constructs Tulczyjew-like discrete triples and Dirac differentials to derive -discrete Lagrange--Dirac equations via -discrete Lagrange--d'Alembert--Pontryagin principles. The authors show equivalence of the resulting equations with established discrete Lagrange--d'Alembert equations, ensure preservation of the -discrete induced Dirac structures in nonholonomic integrators, and validate the approach with numerical tests on a vertical rolling disk and a classical Heisenberg system. The work provides a robust, structure-preserving discretization strategy for nonholonomic mechanics and lays groundwork for extensions to discrete Hamilton–Dirac systems and higher-dimensional/thermodynamic settings. Overall, it offers a unified geometric and variational pathway to accurately simulate constrained mechanical systems in a discrete setting.

Abstract

In this paper, we propose the concept of -discrete Dirac structures over a manifold, where we define -discrete two-forms on the manifold and incorporate discrete constraints using -finite difference maps. Specifically, we develop -discrete induced Dirac structures as discrete analogues of the induced Dirac structure on the cotangent bundle over a configuration manifold, as described by Yoshimura and Marsden (2006). We demonstrate that -discrete Lagrange--Dirac systems can be naturally formulated in conjunction with the -induced Dirac structure on the cotangent bundle. Furthermore, we show that the resulting equations of motion are equivalent to the -discrete Lagrange--d'Alembert equations proposed in Cortés and Martínez (2001) and McLachlan and Perlmutter (2006). We also clarify the variational structures of the discrete Lagrange--Dirac dynamical systems within the framework of the -discrete Lagrange--d'Alembert--Pontryagin principle. Finally, we validate the proposed discrete Lagrange--Dirac systems with some illustrative examples of nonholonomic systems through numerical tests.

Paper Structure

This paper contains 93 sections, 5 theorems, 247 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Variational integrators and nonholonomic integrators.
  3. Dirac structures in mechanics.
  4. Variational structures of Dirac dynamical systems.
  5. Discretization of Dirac structures.
  6. Our goals and the organization of this Paper.
  7. Variational formulation in Lagrangian mechanics
  8. Hamilton's variational principle
  9. The path space and variations of curves.
  10. Hamilton's variational principle.
  11. The second-order vector fields.
  12. The Legendre transform.
  13. Energy conservation.
  14. Canonical forms on the cotangent bundle.
  15. Lagrangian forms on the tangent bundle.
  16. ...and 78 more sections

Key Result

Theorem 5.7

Let $\Omega_{M}^{d\pm}$ be the discrete two-form given in Def. DiscTwoForm_M and let $\Delta_{M}^{d\pm}$ be the discrete constraint spaces given in Def. DiscConstraintSpace_M. Then, the $(+)$-discrete structure $D^{d+}_{M} \subset (M\times M)\times T^{\ast}M$ that is defined by, for each $x_{1} \in is a Dirac structure on $M$. Further, the $(-)$-discrete structure $D^{d-}_{M} \subset (M\times M)\

Figures (20)

  • Figure 1: Variations $\delta q(t)$ of a curve $q(t)$.
  • Figure 2: Discrete sequences.
  • Figure 3: The bundle picture of the iterated tangent and cotangent bundles.
  • Figure 4: $(+)$-discrete bundle picture.
  • Figure 5: $(-)$-discrete bundle picture.
  • ...and 15 more figures

Theorems & Definitions (40)

  • Definition 3.1
  • Remark 4.1
  • Definition 4.2
  • Definition 5.1
  • Remark 5.2
  • Definition 5.3
  • Definition 5.4
  • Definition 5.5
  • Definition 5.6
  • Theorem 5.7
  • ...and 30 more