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Discretization of Dirac systems and port-Hamiltonian systems: the role of the constraint algorithm

María Barbero-Liñán, Juan Manuel López Medel, David Martín de Diego

TL;DR

This work develops a geometry-aware discretization framework for Dirac and port-Hamiltonian systems using retraction and discretization maps, along with their cotangent lifts, to construct structure-preserving integrators. It analyzes two discretization strategies: discretize first then apply the constraint algorithm, and apply the constraint algorithm first and then discretize on the integrable subset, illustrating both with point vortices and port-Hamiltonian models. The approach yields symplectic or presymplectic preservation and exact constraint satisfaction in nonholonomic cases, offering practical numerical schemes for complex constrained dynamics. The results advance geometric integration for Dirac-type systems and provide versatile tools for simulating interconnected physical systems with energy and constraint compatibility in a principled way.

Abstract

We study the discretization of (almost-)Dirac structures using the notion of retraction and discretization maps on manifolds. Additionally, we apply the proposed discretization techniques to obtain numerical integrators for port-Hamiltonian systems and we discuss how to merge the discretization procedure and the constraint algorithm associated to systems of implicit differential equations.

Discretization of Dirac systems and port-Hamiltonian systems: the role of the constraint algorithm

TL;DR

This work develops a geometry-aware discretization framework for Dirac and port-Hamiltonian systems using retraction and discretization maps, along with their cotangent lifts, to construct structure-preserving integrators. It analyzes two discretization strategies: discretize first then apply the constraint algorithm, and apply the constraint algorithm first and then discretize on the integrable subset, illustrating both with point vortices and port-Hamiltonian models. The approach yields symplectic or presymplectic preservation and exact constraint satisfaction in nonholonomic cases, offering practical numerical schemes for complex constrained dynamics. The results advance geometric integration for Dirac-type systems and provide versatile tools for simulating interconnected physical systems with energy and constraint compatibility in a principled way.

Abstract

We study the discretization of (almost-)Dirac structures using the notion of retraction and discretization maps on manifolds. Additionally, we apply the proposed discretization techniques to obtain numerical integrators for port-Hamiltonian systems and we discuss how to merge the discretization procedure and the constraint algorithm associated to systems of implicit differential equations.
Paper Structure (24 sections, 10 theorems, 122 equations, 3 figures)

This paper contains 24 sections, 10 theorems, 122 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Dirac structures
  3. Linear Dirac structures
  4. Dirac structures on a manifold
  5. Dirac systems
  6. The constraint algorithm
  7. Retraction maps
  8. Discretization maps
  9. Cotangent lift of discretization maps
  10. Discretization of Dirac structures
  11. Discretization of Dirac systems
  12. Discretization of a Lagrangian system
  13. Two discretization methods for Dirac systems defined by a Hamiltonian a function
  14. A paradigmatic example: Point vortices
  15. Method 1: First discretization
  16. ...and 9 more sections

Key Result

Proposition 2.3

Let $D$ be a Dirac structure on $V$. Define the subspace $F_D\subset V$ to be the projection of $D$ on $V$. Let $\omega_D$ be the 2-form on $F_D$ given by $\omega_D (u,v)=\alpha(v)$, where $u\oplus \alpha\in D$. Then $\omega_D$ is a skew-symmetric form on $F_D$. Conversely, given a vector space $V$, is the only Dirac structure $D$ on $V$ such that $F_D=F$ and $\omega_D=\omega$.

Figures (3)

  • Figure 1: Trajectories of four point vortices obtained with the three numerical methods.
  • Figure 2: Comparison of energy conservation between each of the symplectic method and RK2.
  • Figure 3: Energy conservation of the symplectic Methods 1 and 2.

Theorems & Definitions (36)

  • Definition 2.1
  • Example 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Definition 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • ...and 26 more