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arxiv: 2507.18573 · v2 · submitted 2025-07-24 · 🧮 math.DG · cs.NA· math-ph· math.MP· math.NA· math.SG

Jacobi Hamiltonian Integrators

Ad\'erito Ara\'ujo , Gon\c{c}alo Inoc\^encio Oliveira , Jo\~ao Nuno Mestre This is my paper

Pith reviewed 2026-05-19 02:27 UTC · model grok-4.3

classification 🧮 math.DG cs.NAmath-phmath.MPmath.NAmath.SG
keywords Jacobi manifoldsHamiltonian systemsstructure-preserving integratorsPoisson geometrygeometric integrationdissipative systemshomogeneous Poisson manifolds
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The pith

A method builds structure-preserving integrators for Hamiltonian systems on Jacobi manifolds by mapping them to homogeneous Poisson manifolds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a construction for numerical integrators that maintain the geometric features of Hamiltonian dynamics when the underlying space is a Jacobi manifold. Jacobi manifolds extend Poisson and contact geometry to cover systems that depend on time or include dissipation. The construction proceeds by using the established link between Jacobi structures and homogeneous Poisson structures to adapt earlier integrator methods. If the approach succeeds, it supplies reliable long-time simulations for models in which energy is not conserved and time appears explicitly in the equations.

Core claim

By exploring the correspondence between Jacobi and homogeneous Poisson manifolds, the authors develop the theoretical tools and outline a numerical integration technique for Jacobi Hamiltonian systems that preserves the homogeneity structure.

What carries the argument

The correspondence between Jacobi manifolds and homogeneous Poisson manifolds, which extends integrator techniques while keeping the homogeneity structure intact.

Load-bearing premise

The correspondence between Jacobi manifolds and homogeneous Poisson manifolds can be leveraged to extend integrator techniques while preserving the homogeneity structure.

What would settle it

Numerical integration of a simple Jacobi Hamiltonian system in which a key structural invariant drifts noticeably over many steps.

read the original abstract

We develop a method of constructing structure-preserving integrators for Hamiltonian systems in Jacobi manifolds. Hamiltonian mechanics, rooted in symplectic and Poisson geometry, has long provided a foundation for modeling conservative systems in classical physics. Jacobi manifolds, generalizing both contact and Poisson manifolds, extend this theory and are suitable for incorporating time-dependent, dissipative and thermodynamic phenomena. Building on recent advances in geometric integrators - specifically Poisson Hamiltonian Integrators (PHI), which preserve key features of Poisson systems - we propose a construction of Jacobi Hamiltonian Integrators. Our approach explores the correspondence between Jacobi and homogeneous Poisson manifolds, with the aim of extending the PHI techniques while ensuring preservation of the homogeneity structure. This work develops the theoretical tools required for this generalization and outlines a numerical integration technique compatible with Jacobi dynamics. { By focusing on the homogeneous Poisson perspective instead of direct contact realizations, we establish a clear pathway for constructing structure-preserving integrators for time-dependent and dissipative systems that are embedded in the Jacobi framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript develops a method for constructing structure-preserving integrators for Hamiltonian systems on Jacobi manifolds. It leverages the algebraic correspondence between Jacobi manifolds and homogeneous Poisson manifolds on the cone, lifts the system, applies Poisson Hamiltonian Integrators (PHI) in the lifted setting, and projects the discrete flow back while aiming to preserve homogeneity and the Jacobi bracket. The approach is positioned as a pathway for geometric integration of time-dependent and dissipative systems without relying on direct contact realizations.

Significance. If the central construction holds, the work would provide a systematic extension of PHI techniques to the Jacobi setting, enabling structure-preserving discretizations for a broader class of systems that include dissipation and explicit time dependence. The emphasis on homogeneity preservation and the homogeneous Poisson perspective is a clear strength, as it builds directly on established manifold correspondences and prior PHI results rather than introducing ad-hoc modifications.

major comments (1)
  1. [Outline of numerical integration technique] The outline of the numerical integration technique does not contain an explicit verification that the projected discrete flow preserves the Jacobi bracket (or the contact form in the contact case) up to the integrator's order. In particular, it is not shown that the homogeneity constraint is maintained under the numerical step in a manner that prevents drift upon descent to the original Jacobi manifold; the algebraic correspondence is clear, but the discrete projection step requires a direct check that the Jacobi identity is respected.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report. We appreciate the referee's detailed feedback on our manuscript concerning Jacobi Hamiltonian Integrators. Below, we provide a point-by-point response to the major comment and indicate the revisions we intend to make.

read point-by-point responses

  1. Referee: [Outline of numerical integration technique] The outline of the numerical integration technique does not contain an explicit verification that the projected discrete flow preserves the Jacobi bracket (or the contact form in the contact case) up to the integrator's order. In particular, it is not shown that the homogeneity constraint is maintained under the numerical step in a manner that prevents drift upon descent to the original Jacobi manifold; the algebraic correspondence is clear, but the discrete projection step requires a direct check that the Jacobi identity is respected.

    Authors: We thank the referee for this observation. The manuscript currently outlines the numerical integration technique by lifting the Jacobi system to a homogeneous Poisson manifold, applying a Poisson Hamiltonian Integrator there, and projecting the discrete flow back, with the aim of preserving the Jacobi structure via the algebraic correspondence. However, we agree that an explicit verification is required to confirm that the projected discrete flow preserves the Jacobi bracket (and contact form where applicable) up to the integrator order, and that the homogeneity constraint is maintained without introducing drift upon descent. In the revised manuscript we will add a dedicated subsection containing this direct check, including a proof that the Jacobi identity is respected by the projected step and a specific treatment of the contact case to rule out drift. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses established correspondence without reduction to inputs

full rationale

The paper's central pathway—lifting Jacobi systems to homogeneous Poisson manifolds, applying PHI integrators, and descending—relies on the algebraic correspondence between Jacobi and homogeneous Poisson structures, which is presented as an external geometric fact rather than a self-definition or fitted input. No equations or steps in the abstract or outline equate the projected discrete flow to the input PHI map by construction, nor do they rename a known result or smuggle an ansatz via self-citation. The construction retains independent theoretical content for preserving homogeneity and Jacobi structure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the established geometric correspondence between Jacobi and homogeneous Poisson manifolds together with the background properties of Jacobi manifolds; no new free parameters or invented entities are introduced in the abstract description.

axioms (1)
  • domain assumption Jacobi manifolds generalize contact and Poisson manifolds and are suitable for incorporating time-dependent, dissipative and thermodynamic phenomena.
    Stated directly in the abstract as the motivation and setting for the work.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Local Universal Splitting Integrators for Contact Hamiltonian Systems

    math.DG 2026-05 unverdicted novelty 7.0

    Proves that the Lie algebra generated by strict and prolonged Hamiltonians is dense in the space of smooth contact Hamiltonians, yielding local universal splitting integrators realized via lifted symplectic and ODE methods.

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