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Jacobi Hamiltonian Integrators: construction and applications

Adérito Araújo, Gonçalo Inocêncio Oliveira, João Nuno Mestre

TL;DR

The paper addresses extending geometric integration to Jacobi manifolds by developing Jacobi Hamiltonian integrators (JHIs). It introduces a systematic framework that uses Poissonization and homogeneous symplectic bi-realizations to lift Jacobi dynamics to homogeneous Poisson systems and applies Magnus-based, order-k constructions via generating functions of Lagrangian bisections. The authors provide backward error analysis, demonstrate energy-dissipation interpretation, and prove preservation of homogeneity and Jacobi invariants, with extensive numerical experiments across contact, low-dimensional, and classical models showing improved qualitative fidelity over standard schemes. The work establishes JHIs as a natural extension of geometric integrators to Jacobi settings, enabling accurate long-time simulations of nonconservative and time-dependent Hamiltonian dynamics and opening avenues for adaptive time-stepping and higher-dimensional applications.

Abstract

We propose a systematic framework for constructing geometric integrators for Hamiltonian systems on Jacobi manifolds. By combining Poissonization of Jacobi structures with homogeneous symplectic bi-realizations, Jacobi dynamics are lifted to homogeneous Poisson Hamiltonian systems, enabling the construction of structure-preserving Jacobi Hamiltonian integrators. The resulting schemes are constructed explicitly and applied to a range of examples, including contact Hamiltonian systems and classical models. Numerical experiments highlight their qualitative advantages over standard integrators, including better preservation of geometric structure and improved long-time behavior.

Jacobi Hamiltonian Integrators: construction and applications

TL;DR

The paper addresses extending geometric integration to Jacobi manifolds by developing Jacobi Hamiltonian integrators (JHIs). It introduces a systematic framework that uses Poissonization and homogeneous symplectic bi-realizations to lift Jacobi dynamics to homogeneous Poisson systems and applies Magnus-based, order-k constructions via generating functions of Lagrangian bisections. The authors provide backward error analysis, demonstrate energy-dissipation interpretation, and prove preservation of homogeneity and Jacobi invariants, with extensive numerical experiments across contact, low-dimensional, and classical models showing improved qualitative fidelity over standard schemes. The work establishes JHIs as a natural extension of geometric integrators to Jacobi settings, enabling accurate long-time simulations of nonconservative and time-dependent Hamiltonian dynamics and opening avenues for adaptive time-stepping and higher-dimensional applications.

Abstract

We propose a systematic framework for constructing geometric integrators for Hamiltonian systems on Jacobi manifolds. By combining Poissonization of Jacobi structures with homogeneous symplectic bi-realizations, Jacobi dynamics are lifted to homogeneous Poisson Hamiltonian systems, enabling the construction of structure-preserving Jacobi Hamiltonian integrators. The resulting schemes are constructed explicitly and applied to a range of examples, including contact Hamiltonian systems and classical models. Numerical experiments highlight their qualitative advantages over standard integrators, including better preservation of geometric structure and improved long-time behavior.
Paper Structure (33 sections, 4 theorems, 104 equations, 18 figures, 5 tables)

This paper contains 33 sections, 4 theorems, 104 equations, 18 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Construction of Jacobi Hamiltonian integrators
  3. Construction of the method
  4. Poissonization of Jacobi Hamiltonian systems
  5. Homogeneous symplectic bi-realization
  6. Iterative construction of the method
  7. Structure-preserving properties
  8. Magnus expansion
  9. Backward error analysis
  10. Dissipation of energy in Jacobi Hamiltonian systems
  11. Homogeneity of Lagrangian bisections
  12. Numerical examples
  13. Step 1: Poissonization.
  14. Step 2: Symplectic bi-realization.
  15. Step 3: Computation of $S_i$.
  16. ...and 18 more sections

Key Result

Theorem 2.1

The recursion defined in steps 1 and 2 preserves the homogeneous structure; that is, all generated $S_i$ are 1-homogeneous.

Figures (18)

  • Figure 1: Flowchart for the construction of Jacobi Hamiltonian Integrators (JHI).
  • Figure 2: Comparison between JHI and RK-2 with the exact solution for the contact Hamiltonian system.
  • Figure 3: Comparison between the exact trajectory and those obtained by the JHI and RK-2 methods for the Hamiltonian $H=x^2+y^2$.
  • Figure 4: Comparison between the exact trajectory and those obtained by the JHI and RK-2 methods for the Hamiltonian $H=\cos(x)\sin(y)$.
  • Figure 5: Hamiltonian error along trajectories computed with JHI and RK-2.
  • ...and 13 more figures

Theorems & Definitions (14)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3: Magnus Magnus1954
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Theorem 2.9: COSSERAT2023
  • ...and 4 more