Jacobi Hamiltonian Integrators

• Jacobi Hamiltonian integrators are geometric numerical methods that preserve the Jacobi structure by lifting dynamics to a homogeneous Poisson manifold.
• They utilize Poissonization, homogeneous symplectic bi-realizations, and Lagrangian bisections to construct accurate discrete flows.
• These integrators enable robust simulation of contact, time-dependent, and dissipative systems while improving conservation of Casimirs and energy behavior.

Jacobi Hamiltonian integrators are geometric numerical methods for Hamiltonian systems on Jacobi manifolds. Their defining construction lifts Jacobi dynamics to a homogeneous Poisson manifold by Poissonization, builds the discrete flow in that homogeneous Poisson setting through homogeneous symplectic bi-realizations, Lagrangian bisections, and Hamilton–Jacobi data, and then projects the resulting map back to the original Jacobi manifold. In this way, the discrete evolution is designed to preserve the Jacobi structure rather than merely approximate trajectories, with particular relevance for contact, time-dependent, dissipative, and thermodynamic models (Araújo et al., 24 Jul 2025, Araújo et al., 28 Jan 2026).

1. Geometric setting: Jacobi manifolds and Hamiltonian dynamics

A Jacobi manifold is a smooth manifold $J$ equipped with a bivector field $\Lambda\in\mathfrak{X}^2(J)$ and a vector field $E\in\mathfrak{X}(J)$ satisfying

$$[\Lambda,\Lambda]=2E\wedge \Lambda,\qquad [\Lambda,E]=0,$$

with $[\cdot,\cdot]$ the Schouten–Nijenhuis bracket. From this pair one obtains the Jacobi bracket

$$\{f_1,f_2\}_J = \Lambda(df_1,df_2)+f_1E(f_2)-f_2E(f_1), \qquad f_1,f_2\in C^\infty(J),$$

which is a Lie bracket. Unlike the Poisson bracket, it is not a derivation in each argument in the usual sense because of the extra $E$-terms (Araújo et al., 24 Jul 2025).

Given a Hamiltonian $H\in C^\infty(J)$, the associated Jacobi Hamiltonian vector field is

$X_H=\Lambda(\cdot,dH)-H\,E(\cdot),$

equivalently the derivation $f\mapsto \{H,f\}_J$. In particular, $\Lambda\in\mathfrak{X}^2(J)$0 itself is the Hamiltonian vector field of the constant function $\Lambda\in\mathfrak{X}^2(J)$1 (Araújo et al., 24 Jul 2025). The formulation includes Poisson manifolds as the special case $\Lambda\in\mathfrak{X}^2(J)$2, $\Lambda\in\mathfrak{X}^2(J)$3, and contact manifolds as Jacobi manifolds. For a contact manifold with local Darboux coordinates $\Lambda\in\mathfrak{X}^2(J)$4, the canonical Jacobi structure is

$\Lambda\in\mathfrak{X}^2(J)$5

The motivation for integrators in this setting is that Jacobi Hamiltonian vector fields naturally encode non-conservation of the Hamiltonian in a controlled way, and many systems of interest are not purely symplectic or Poisson in the classical conservative sense. Contact Hamiltonian systems model dissipation, friction, and thermodynamics, while Jacobi manifolds unify Poisson and contact geometry (Araújo et al., 28 Jan 2026).

2. Poissonization and the homogeneous Poisson viewpoint

The central reduction underlying Jacobi Hamiltonian integrators is Poissonization. Starting from $\Lambda\in\mathfrak{X}^2(J)$6, one considers the principal $\Lambda\in\mathfrak{X}^2(J)$7-bundle

$\Lambda\in\mathfrak{X}^2(J)$8

with action

$\Lambda\in\mathfrak{X}^2(J)$9

On $E\in\mathfrak{X}(J)$0, the Jacobi structure becomes the homogeneous Poisson structure

$E\in\mathfrak{X}(J)$1

with Euler vector field

$E\in\mathfrak{X}(J)$2

The homogeneity condition is

$E\in\mathfrak{X}(J)$3

so $E\in\mathfrak{X}(J)$4 is $E\in\mathfrak{X}(J)$5-homogeneous (Araújo et al., 24 Jul 2025).

This homogeneous Poisson description is not auxiliary; it is the mechanism by which Jacobi dynamics become accessible to Poisson and symplectic integration technology. The papers state the categorical equivalence

$E\in\mathfrak{X}(J)$6

and, similarly, the equivalence between contact and homogeneous symplectic geometry in the appropriate setting (Araújo et al., 24 Jul 2025).

For Hamiltonian dynamics, the lifted homogeneous Hamiltonian is

$E\in\mathfrak{X}(J)$7

Its Hamiltonian vector field on the Poissonized space satisfies

$E\in\mathfrak{X}(J)$8

so the original Jacobi flow is the projection of a homogeneous Poisson Hamiltonian flow. A crucial identity is

$E\in\mathfrak{X}(J)$9

which means that the lifted system conserves the homogeneous Hamiltonian $$[\Lambda,\Lambda]=2E\wedge \Lambda,\qquad [\Lambda,E]=0,$$0, even though $$[\Lambda,\Lambda]=2E\wedge \Lambda,\qquad [\Lambda,E]=0,$$1 itself may vary along Jacobi trajectories (Araújo et al., 28 Jan 2026).

Poissonization also transports invariant objects. A Jacobi-Casimir $$[\Lambda,\Lambda]=2E\wedge \Lambda,\qquad [\Lambda,E]=0,$$2 satisfies

$$[\Lambda,\Lambda]=2E\wedge \Lambda,\qquad [\Lambda,E]=0,$$3

Viewed as a $$[\Lambda,\Lambda]=2E\wedge \Lambda,\qquad [\Lambda,E]=0,$$4-homogeneous function on $$[\Lambda,\Lambda]=2E\wedge \Lambda,\qquad [\Lambda,E]=0,$$5, it satisfies

$$[\Lambda,\Lambda]=2E\wedge \Lambda,\qquad [\Lambda,E]=0,$$6

if and only if it is a Jacobi-Casimir. Thus Jacobi-Casimirs become homogeneous Poisson-Casimirs (Araújo et al., 24 Jul 2025).

3. Construction of Jacobi Hamiltonian integrators

The geometric construction proceeds by adapting Poisson Hamiltonian integrator machinery to the homogeneous setting. The strategy is: Poissonize the Jacobi system, construct a homogeneous symplectic bi-realization of the homogeneous Poisson manifold, approximate the Hamiltonian flow by homogeneous Lagrangian bisections, and project the induced homogeneous Poisson map back to a Jacobi map on $$[\Lambda,\Lambda]=2E\wedge \Lambda,\qquad [\Lambda,E]=0,$$7 (Araújo et al., 24 Jul 2025).

A symplectic realization of a Poisson manifold $$[\Lambda,\Lambda]=2E\wedge \Lambda,\qquad [\Lambda,E]=0,$$8 is a surjective Poisson submersion

$$[\Lambda,\Lambda]=2E\wedge \Lambda,\qquad [\Lambda,E]=0,$$9

and a bi-realization is a pair $[\cdot,\cdot]$0 such that $[\cdot,\cdot]$1 is Poisson, $[\cdot,\cdot]$2 is anti-Poisson, and source and target fibers are symplectically orthogonal. In the homogeneous setting, $[\cdot,\cdot]$3 and $[\cdot,\cdot]$4 carry compatible $[\cdot,\cdot]$5-actions and all relevant maps and forms are homogeneous (Araújo et al., 24 Jul 2025).

The realization can be constructed from a homogeneous Poisson spray $[\cdot,\cdot]$6, satisfying

$[\cdot,\cdot]$7

For the Poissonization $[\cdot,\cdot]$8, the spray is written explicitly in coordinates, and from its flow $[\cdot,\cdot]$9 one defines

$$\{f_1,f_2\}_J = \Lambda(df_1,df_2)+f_1E(f_2)-f_2E(f_1), \qquad f_1,f_2\in C^\infty(J),$$0

together with the symplectic form

$$\{f_1,f_2\}_J = \Lambda(df_1,df_2)+f_1E(f_2)-f_2E(f_1), \qquad f_1,f_2\in C^\infty(J),$$1

Because everything is homogeneous, $$\{f_1,f_2\}_J = \Lambda(df_1,df_2)+f_1E(f_2)-f_2E(f_1), \qquad f_1,f_2\in C^\infty(J),$$2 is $$\{f_1,f_2\}_J = \Lambda(df_1,df_2)+f_1E(f_2)-f_2E(f_1), \qquad f_1,f_2\in C^\infty(J),$$3-homogeneous. The resulting statement is that any homogeneous Poisson spray induces a homogeneous bi-realization (Araújo et al., 24 Jul 2025).

The dynamical step is encoded by homogeneous Lagrangian bisections. For a homogeneous symplectic bi-realization $$\{f_1,f_2\}_J = \Lambda(df_1,df_2)+f_1E(f_2)-f_2E(f_1), \qquad f_1,f_2\in C^\infty(J),$$4, a Lagrangian bisection $$\{f_1,f_2\}_J = \Lambda(df_1,df_2)+f_1E(f_2)-f_2E(f_1), \qquad f_1,f_2\in C^\infty(J),$$5 induces

$$\{f_1,f_2\}_J = \Lambda(df_1,df_2)+f_1E(f_2)-f_2E(f_1), \qquad f_1,f_2\in C^\infty(J),$$6

If $$\{f_1,f_2\}_J = \Lambda(df_1,df_2)+f_1E(f_2)-f_2E(f_1), \qquad f_1,f_2\in C^\infty(J),$$7 is Lagrangian, then $$\{f_1,f_2\}_J = \Lambda(df_1,df_2)+f_1E(f_2)-f_2E(f_1), \qquad f_1,f_2\in C^\infty(J),$$8 is a Poisson diffeomorphism; if $$\{f_1,f_2\}_J = \Lambda(df_1,df_2)+f_1E(f_2)-f_2E(f_1), \qquad f_1,f_2\in C^\infty(J),$$9 is homogeneous, then $E$0 is a homogeneous Poisson diffeomorphism. For an exact family $E$1 with $E$2 the zero section, the discrete time-$E$3 update is

$E$4

Moreover, the exact Hamiltonian flow is recovered from the evolved Lagrangian bisection,

$E$5

so approximating the bisection $E$6 produces an approximation to the Hamiltonian flow (Araújo et al., 24 Jul 2025).

The homogeneous Hamilton–Jacobi side supplies the generating data. For a $E$7-homogeneous generating function $E$8,

$E$9

and choosing $H\in C^\infty(J)$0 gives the homogeneous Hamilton–Jacobi equation

$H\in C^\infty(J)$1

In the more explicit recursive formulation, one sets

$H\in C^\infty(J)$2

and then defines

$H\in C^\infty(J)$3

A key theorem is that this recursion preserves homogeneity: if $H\in C^\infty(J)$4 are $H\in C^\infty(J)$5-homogeneous, then $H\in C^\infty(J)$6 is $H\in C^\infty(J)$7-homogeneous (Araújo et al., 28 Jan 2026).

With step size $H\in C^\infty(J)$8, the order-$H\in C^\infty(J)$9 Jacobi Hamiltonian integrator is then

$X_H=\Lambda(\cdot,dH)-H\,E(\cdot),$0

followed by

$X_H=\Lambda(\cdot,dH)-H\,E(\cdot),$1

For first order, only $X_H=\Lambda(\cdot,dH)-H\,E(\cdot),$2 is retained. The papers also state that the Magnus expansion is used to organize the higher-order terms and to justify backward-error analysis, yielding a modified homogeneous Hamiltonian $X_H=\Lambda(\cdot,dH)-H\,E(\cdot),$3 such that

$X_H=\Lambda(\cdot,dH)-H\,E(\cdot),$4

for the discrete flow (Araújo et al., 28 Jan 2026).

4. Relation to Poisson integrators, generating functions, and groupoid methods

Jacobi Hamiltonian integrators extend a broader program in geometric integration that emphasizes preservation of reduced phase-space geometry rather than only local truncation error. For Poisson systems

$X_H=\Lambda(\cdot,dH)-H\,E(\cdot),$5

standard methods generally fail to preserve the Poisson tensor, symplectic leaves, Casimirs, or exact energy when $X_H=\Lambda(\cdot,dH)-H\,E(\cdot),$6 is variable; even the midpoint rule is Poisson only when $X_H=\Lambda(\cdot,dH)-H\,E(\cdot),$7 is constant, and otherwise is only an “almost-Poisson integrator,” preserving the bracket up to $X_H=\Lambda(\cdot,dH)-H\,E(\cdot),$8. The corresponding geometric requirement is that a numerical method should be viewed as an approximation of a Poisson map or canonical relation, not merely as a solver for differential equations (Diego, 2018).

The JHI framework is presented as a homogeneous version of Poisson Hamiltonian Integrators. In the Poisson case, Hamiltonian dynamics on a Poisson manifold can be encoded through local symplectic groupoids or bi-realizations, Hamilton–Jacobi equations, and Lagrangian bisections; the homogeneous Jacobi construction keeps that architecture while enforcing compatibility with the $X_H=\Lambda(\cdot,dH)-H\,E(\cdot),$9-action (Araújo et al., 24 Jul 2025).

This places JHIs within a generating-function lineage that is also visible in discrete Hamiltonian variational integrators. There, the exact discrete right Hamiltonian is the exact Type II generating function for the time-$f\mapsto \{H,f\}_J$0 Hamiltonian flow,

$f\mapsto \{H,f\}_J$1

and the discrete Hamiltonian map is defined directly by derivatives of a discrete generating function (Leok et al., 2010). A plausible implication is that JHIs inherit this same emphasis on generating objects—here homogeneous generating functions and homogeneous Lagrangian bisections—rather than direct time discretization of the base ODE.

The groupoid formulation has also acquired an explicit Butcher-series and Hopf-algebraic expression. In the symplectic groupoid setting, Hamilton–Jacobi solutions and Lagrangian bisections can be approximated through a pre-Lie algebra of rooted-tree type, with B-series and the Butcher–Connes–Kreimer coproduct encoding the composition law of bisections. The resulting Taylor-Hamiltonian-Poisson and Runge-Kutta-Hamiltonian-Poisson schemes are Poisson integrators by construction (Laurent et al., 6 Mar 2025). JHIs use the same symplectic-groupoid logic, but with the additional homogeneity constraint needed for descent from the Poissonized space back to a Jacobi manifold (Araújo et al., 24 Jul 2025).

5. Applications and numerical behavior

The first JHI paper is mainly theoretical and outlines a numerical integration technique compatible with Jacobi dynamics, whereas the later construction-and-applications paper gives explicit schemes and numerical experiments (Araújo et al., 24 Jul 2025, Araújo et al., 28 Jan 2026). The tested examples include contact Hamiltonian systems and several low-dimensional Jacobi models.

For the canonical contact structure in coordinates $f\mapsto \{H,f\}_J$2,

$f\mapsto \{H,f\}_J$3

the Jacobi Hamiltonian vector field is

$f\mapsto \{H,f\}_J$4

For the damped harmonic oscillator

$f\mapsto \{H,f\}_J$5

first- and third-order JHIs are constructed. The third-order term contains

$f\mapsto \{H,f\}_J$6

The reported behavior is that JHI-1 already outperforms a semi-implicit symplectic Euler method, JHI-3 nearly overlaps with the exact solution, and Hamiltonian deviation remains small and behaves consistently with the expected dissipation (Araújo et al., 28 Jan 2026).

In the two-dimensional Jacobi example,

$f\mapsto \{H,f\}_J$7

the Hamiltonian

$f\mapsto \{H,f\}_J$8

leads to an exact method in the sense that $f\mapsto \{H,f\}_J$9 for $\Lambda\in\mathfrak{X}^2(J)$00. For

$\Lambda\in\mathfrak{X}^2(J)$01

the first-order JHI exhibits near second-order convergence and better Hamiltonian behavior than RK-2 (Araújo et al., 28 Jan 2026).

In the three-dimensional example,

$\Lambda\in\mathfrak{X}^2(J)$02

with Hamiltonian

$\Lambda\in\mathfrak{X}^2(J)$03

the Poissonized system has Casimirs

$\Lambda\in\mathfrak{X}^2(J)$04

The reported numerical behavior is especially strong: JHI-1 gives second-order convergence, JHI-3 gives fourth-order convergence, the Casimirs are preserved to machine precision, and the lifted Hamiltonian is much better preserved than by RK-2 (Araújo et al., 28 Jan 2026).

The four-dimensional example uses

$\Lambda\in\mathfrak{X}^2(J)$05

Because a full exact symplectic realization is difficult, the construction uses an approximate first-order homogeneous bi-realization. Even so, the method shows convergence close to second order, small Hamiltonian and Casimir errors, and robust qualitative agreement with exact trajectories (Araújo et al., 28 Jan 2026).

The modified Lotka–Volterra example is reformulated as a Jacobi system with

$\Lambda\in\mathfrak{X}^2(J)$06

and Hamiltonian

$\Lambda\in\mathfrak{X}^2(J)$07

The numerical findings are that JHI-1 preserves the closed-orbit structure much better than RK-2, Hamiltonian error remains periodic and bounded, and long-time qualitative dynamics are superior (Araújo et al., 28 Jan 2026).

A rigid-body rotation example employs a linear Poisson-type $\Lambda\in\mathfrak{X}^2(J)$08 and a quadratic $\Lambda\in\mathfrak{X}^2(J)$09. Since an exact symplectic bi-realization is hard to obtain, an approximate bi-realization preserving the canonical Poisson brackets to first order is used. For certain inertial parameters the JHI trajectory coincides with the exact one; in more unstable cases it preserves closed behavior, though some deviation appears because the bi-realization is only approximate; Hamiltonian errors stay bounded (Araújo et al., 28 Jan 2026).

Across these tests, the reported pattern is that JHIs preserve the Jacobi structure, improve preservation of Casimir functions, reduce drift in the lifted homogeneous Hamiltonian, better reproduce closed orbits and phase portrait features, and handle contact dissipation correctly. The paper also reports that JHI-1 often converges with observed order $\Lambda\in\mathfrak{X}^2(J)$10, and JHI-3 often with observed order $\Lambda\in\mathfrak{X}^2(J)$11 (Araújo et al., 28 Jan 2026).

6. Scope, limitations, and terminological distinctions

The JHI framework is explicitly geometric and structure-preserving, but its present state includes significant limitations. The initial paper is mainly a theoretical development: it does not yet present a fully implemented, benchmarked numerical method; explicit realization maps may be hard to compute exactly, so approximations are needed; the treatment is given in a trivial line bundle setting, not the full generality of Jacobi structures on arbitrary line bundles; and global existence is not guaranteed, since local symplectic groupoids or realizations are used (Araújo et al., 24 Jul 2025). The subsequent paper addresses some of these issues by constructing explicit schemes and applying them to several examples, but exact or approximate homogeneous symplectic bi-realizations remain central technical inputs (Araújo et al., 28 Jan 2026).

The framework also adopts a specific geometric stance. Direct development through contact realizations and local contact groupoids is identified as an alternative route, but the JHI construction deliberately proceeds through homogeneous Poisson geometry because the Poisson Hamiltonian Integrator machinery is naturally Poisson/symplectic and the homogeneity conditions can be checked directly and systematically (Araújo et al., 24 Jul 2025).

A common terminological ambiguity concerns the word “Jacobi.” In Jacobi Hamiltonian integrators, “Jacobi” refers to Jacobi manifolds. The topic is therefore distinct from “Symplectic integration with Jacobi polynomials” (Tang, 2018), where Jacobi polynomials parameterize families of symplectic continuous-stage Runge–Kutta methods for canonical Hamiltonian systems, and from the phase-space “Jacobi algorithm” for diagonalizing Hamiltonian and skew-Hamiltonian matrices by successive elementary symplectic decoupling transformations (Baumgarten, 2020). There is, however, a separate historical use of “Jacobi-style” in discrete Hamiltonian mechanics: discrete Hamiltonian variational integrators are described as the natural discrete analogues of the Type II, Jacobi-style Hamiltonian generating-function picture of classical mechanics (Leok et al., 2010). This suggests a conceptual overlap at the level of generating functions, but not an identity of subject matter.

In its current form, the JHI program defines a systematic route for geometric integration on Jacobi manifolds: Poissonize, impose homogeneity, build a homogeneous symplectic bi-realization, represent the exact flow by homogeneous Lagrangian bisections or recursive generating functions, approximate those objects, and descend to a Jacobi map. Its distinctive claim is that the preservation target is the Jacobi structure itself, together with the homogeneity that encodes it, rather than only the underlying differential equation (Araújo et al., 24 Jul 2025, Araújo et al., 28 Jan 2026).