Linear Hamiltonian Vector Fields on Lie Groups

Abstract

Linear vector fields on Lie groups constitute a fundamental class of dynamical systems, as their flows are one-parameter subgroups of automorphisms and their infinitesimal behavior is entirely determined by derivations of the Lie algebra. When a Lie group is endowed with a Hamiltonian-type geometric structure, a natural problem is to determine whether such linear dynamics admit a global variational realization, and how such realizations can be interpreted in terms of reduced models of fluid motion. In the even-dimensional case, where the Lie group carries a symplectic structure, we establish a complete global criterion for the existence of Hamiltonians generating linear symplectic vector fields. The problem reduces to a single global obstruction: the de Rham cohomology class of the 1-form ${\iota }_{\mathcal{X}}\omega$. Thus, every linear symplectic vector field on a simply connected Lie group is globally Hamiltonian, and when the obstruction vanishes, we provide an explicit constructive procedure to recover the Hamiltonian. On the affine group ${\mathrm{Aff}}^{+}\left(1\right)$, this yields a fully explicit, finite-dimensional Hamiltonian model of a 1D ideal fluid with affine symmetries. We then treat odd-dimensional Lie groups, where symplectic geometry is unavailable. Using contact geometry as the canonical replacement, we prove a Hamiltonian lifting theorem ensuring the existence and uniqueness of the associated dynamics. The Reeb vector field appears as a distinguished vertical direction resolving the ambiguities of degenerate Hamiltonian systems. On the Heisenberg group $H_3$, this gives a fully explicit contact Hamiltonian model of an effective non-conservative fluid mode. Finally, we interpret symplectic and contact theories within a unified geometric framework and discuss their relevance to geometric formulations of ideal (symplectic) and effective (contact) fluid equations on Lie groups.

1. Introduction

Hamiltonian dynamics on Lie groups lies at the intersection of geometric mechanics, symmetry, fluid dynamics, and control theory [1,2,3,4,5,6,7]. The combination of variational principles and group structure yields models that are both mathematically rigorous and practically relevant. Within this setting, linear vector fields play a distinguished role: their flows are one-parameter subgroups of the automorphism group, and their infinitesimal generators are derivations of the Lie algebra. As a result, linear systems on Lie groups admit explicit descriptions in control theory and in almost Riemannian structures, and constitute a natural testing ground for global geometric questions [8,9,10].

From the viewpoint of geometric fluid dynamics in the sense of Arnold and Marsden-Ratiu [2,5], respectively, the Lie groups considered in this work (such as ${\mathrm{Aff}}^{+}\left(1\right)$ and the Heisenberg group $H_3$) can be regarded as finite-dimensional symmetry reductions or collective-mode models of fluid motion. In the symplectic setting, linear Hamiltonian vector fields on $(G,\omega )$ provide idealized descriptions of reduced, energy-conserving flows (“ideal fluid modes”), where global existence of a Hamiltonian can be interpreted as the existence of a well-defined reduced energy functional. In the contact setting [11,12,13], linear contact Hamiltonian vector fields on $(G,\alpha )$ naturally encode effective non-conservative behavior: the Reeb direction plays the role of an internal or entropic variable accumulating dissipation, while the horizontal distribution models the instantaneous mechanical motion.

A central problem addressed in this work is the following: when does a linear vector field preserving a Hamiltonian-type geometric structure admit a globally defined variational realization? While local Hamiltonian descriptions are always available under mild regularity assumptions, global realizability may fail due to topological obstructions. Understanding the precise nature of these obstructions is essential for the consistent use of Hamiltonian methods on Lie groups and for applications in geometric mechanics, geometric fluid dynamics, and control [5,6,7,11,14,15,16,17].

In the even-dimensional case, where the Lie group carries a symplectic structure, the Hamiltonian realizability problem admits a remarkably sharp answer. We show that a linear symplectic vector field is Hamiltonian if and only if a naturally associated closed 1-form is exact. The obstruction is therefore global and purely cohomological, and it vanishes for simply connected groups. This result provides a complete and intrinsic characterization of Hamiltonian linear dynamics in the symplectic category, valid for arbitrary symplectic Lie groups. In particular, on the affine group ${\mathrm{Aff}}^{+}\left(1\right)$ this yields a fully explicit, finite-dimensional Hamiltonian model of a 1D ideal fluid with affine symmetries, where the global scale and translation form a symplectic phase space, and the reduced dynamics are governed by a global energy functional depending only on the homogeneous deformation.

Many Lie groups of fundamental importance are odd-dimensional, where symplectic geometry is no longer available. For instance, in dimension three we mention the Heisenberg nilpotent Lie group, the solvable group $SE\left(2\right)$ of rigid movements of the plane, and the semi-simple groups: the sphere, the rotational compact group $SO(3,R)$ and $SL(2,R)$ the group of matrices of order two and determinant 1. In this setting, the classical Hamiltonian lifting fails and must be replaced by an alternative geometric mechanism. We show that contact geometry provides the correct odd-dimensional analogue, leading to a well-posed Hamiltonian lifting theory with existence and uniqueness. A distinguished vertical direction is given by the Reeb vector field [13]. On the Heisenberg group $H_3$, this contact framework provides a fully explicit, finite-dimensional contact Hamiltonian model of an effective non-conservative fluid mode, in which an internal (Reeb) direction accumulates a quantity proportional to a conserved horizontal energy, thereby offering a simple prototype of a reduced dissipative or entropic fluid behavior in finite dimensions.

In what follows, we will provide the definition of contact manifolds and Reeb Field.

A contact manifold is a geometric structure on a differentiable manifold M of dimension $2n+1$ defined by a hyperplane that carries information about the direction in which a “contact” can be defined. Formally, a $(n+1)$-form $\alpha$ on M is said to define a contact structure if it satisfies the following conditions:

• $\alpha$ is closed:

  $$d\alpha =0.$$
• The $(n+1)$-form ${\alpha }^n\wedge d\alpha$ is non-vanishing at every point of M.

A Reeb field on a contact manifold is a vector field associated with the contact form $\alpha$. It is defined formally as the vector field that satisfies the following conditions:

• $\alpha \left(R\right)=1$: The Reeb field is normalized so that the contact form $\alpha$ evaluates to 1 on it.
• $d\alpha (R,·)=0$: The Reeb field is annihilated by the differential form $d\alpha$.

This vector field plays a fundamental role in contact geometry, describing the “directions of evolution” in the contact space. The paper is organized as follows. In Section 2, we treat the even-dimensional symplectic case and establish the global Hamiltonian criterion for linear vector fields, incorporating all necessary geometric ingredients directly into the proof. Section 3 presents the affine model ${\mathrm{Aff}}^{+}\left(1\right)$ of a 1D ideal fluid with affine symmetries. In Section 4, we develop the odd-dimensional contact Hamiltonian lifting theory, again in a fully self-contained manner, and Section 5 illustrates it through a concrete example on the Heisenberg contact model of an effective non-conservative fluid mode. Section 6 contains a unified discussion of conclusions. We end in Section 7 with a perspective and future work.

2. Linear Hamiltonian Fields on Symplectic Lie Groups

Let G be a connected Lie group endowed with a symplectic form $\omega \in {\mathsf{\Omega }}^2\left(G\right)$. We address the problem of Hamiltonian realizability for linear vector fields on $(G,\omega )$, emphasizing the intrinsic and global nature of the obstruction.

A vector field $\mathcal{X}$ on G is called linear if its flow ${\left\{{\phi }_t\right\}}_{t\in \mathbb{R}}$ is a one-parameter subgroup of $\mathrm{Aut}\left(G\right)$ [9]. Equivalently, $\mathcal{X}$ is generated by a derivation $D\in \mathrm{Der}\left(\mathfrak{g}\right)$, where $\mathfrak{g}$ is the Lie algebra of G, identify with $T_{e}G$, the tangent space of the group via

$${\phi }_t=exp\left(tD\right),\mathcal{X}\left(g\right)={\left.{\displaystyle \frac{d}{dt}}\right|}_{t=0}{\phi }_t\left(g\right).$$

The central question in the symplectic setting is the following: When does a linear vector field on a symplectic Lie group admit a globally defined Hamiltonian function? We now state the main result for the even-dimensional case. Global Hamiltonian criterion for linear symplectic fields.

Theorem 1.

Let $(G,\omega )$ be a symplectic Lie group, where ω is a left invariant 2-form and let $\mathcal{X}$ be a linear vector field on G. Define the 1-form

$$\alpha :={\iota }_{\mathcal{X}}\omega .$$

Then, the following statements are equivalent:

• There exists $H\in C^{\infty }\left(G\right)$ such that

  $${\iota }_{\mathcal{X}}\omega =-dH,$$

  i.e., $\mathcal{X}$ is a Hamiltonian vector field.
• The 1-form α is exact.
• ${\mathcal{L}}_{\mathcal{X}}\omega =0$ and $\left[\alpha \right]=0$ in $H_{\mathrm{dR}}^1\left(G\right)$.

In particular, if G is simply connected, then

$$\mathcal{X}\mathit{is}\mathit{Hamiltonian}⟺{\mathcal{L}}_{\mathcal{X}}\omega =0.$$

Moreover, whenever these conditions hold, a Hamiltonian H generating $\mathcal{X}$ can be constructed explicitly.

Proof. 

The proof is organized in three conceptual steps. All geometric and algebraic ingredients required for the argument are recalled within the proof itself. We start with a symplectic structure and the Hamiltonian vector fields.

Since $\omega$ is symplectic, it is closed and nondegenerate, and therefore induces a vector bundle isomorphism

$${\omega }^{♭}:TG⟶T^{*}G,v⟼{\iota }_v\omega ,$$

defined pointwise by

$${\mathsf{\Omega }}_x^{♭}\left(v\right)\left(w\right)={\mathsf{\Omega }}_x(v,w),v,w\in T_{x}G.$$

Thus, ${\mathsf{\Omega }}_x^{♭}$ is the linear map encoded by the bilinear form ${\mathsf{\Omega }}_x$.

A vector field X on G is Hamiltonian if there exists a smooth function $H\in C^{\infty }\left(G\right)$ such that

$${\iota }_X\omega =-dH.$$

Every Hamiltonian vector field preserves $\omega$. Conversely, a vector field X is said to be symplectic if ${\mathcal{L}}_X\omega =0$. Since $\mathrm{d}\omega =0$, Cartan’s identity yields

$${\mathcal{L}}_X\omega =\mathrm{d}\left({\iota }_X\omega \right),$$

and therefore ${\mathcal{L}}_X\omega =0$ if and only if the 1-form ${\iota }_X\omega$ is closed. Thus, the obstruction to global Hamiltonian realizability is purely cohomological and measured by the class

$$\left[{\iota }_X\omega \right]\in H_{\mathrm{dR}}^1\left(G\right),$$

see the reference [5].

Assume now that $\mathcal{X}$ is linear. Then its flow is a one-parameter subgroup of $\mathrm{Aut}\left(G\right)$ generated by a derivation $D\in \mathrm{Der}\left(\mathfrak{g}\right)$. Let $u^L$ denote the left-invariant vector field associated with $u\in \mathfrak{g}$. Linearity implies

$$[\mathcal{X},u^L]=-{\left(Du\right)}^L.$$

If $\omega$ is left-invariant, its Lie derivative along $\mathcal{X}$ is fully determined by its value at the identity. A direct computation gives

$${\left({\mathcal{L}}_{\mathcal{X}}\omega \right)}_e(u,v)={\omega }_e(Du,v)+{\omega }_e(u,Dv),u,v\in \mathfrak{g}.$$

Therefore,

$${\mathcal{L}}_{\mathcal{X}}\omega =0⟺{\omega }_e(Du,v)+{\omega }_e(u,Dv)=0,\forall u,v\in \mathfrak{g}.$$

This condition characterizes symplectic derivations and is purely algebraic, depending only on the pair $(\mathfrak{g},{\omega }_e)$ [2].

The condition ${\mathcal{L}}_{\mathcal{X}}\omega =0$ implies that $\alpha$ is closed. If $\alpha$ is exact, there exists $H\in C^{\infty }\left(G\right)$ such that $dH=-\alpha$, and therefore $\mathcal{X}=X_H$. This proves the equivalence between (1) and (2).

Statement (3) expresses precisely that $\alpha$ is closed and represents the trivial class in $H_{\mathrm{dR}}^1\left(G\right)$, hence (2) and (3) are equivalent. If G is simply connected, then $H_{\mathrm{dR}}^1\left(G\right)=0$, so every closed 1-form is exact, and every symplectic linear vector field is globally Hamiltonian.

Finally, assume that the 1-form $\alpha$ is closed. To decide whether $\alpha$ is exact, choose a basis ${\left\{{\gamma }_j\right\}}_{j=1}^k$ of the singular homology group $H_1(G;\mathbb{Z})$ and compute the corresponding periods

$${\int }_{{\gamma }_j}\alpha .$$

If at least one of these integrals is nonzero, then $\alpha$ cannot be exact and defines a nontrivial cohomology class

$$\left[\alpha \right]\ne 0\mathrm{in}{H}_{\mathrm{dR}}^1\left(G\right).$$

In this case, there exists no globally defined Hamiltonian function H satisfying $\alpha =-dH$, and therefore the vector field admits no global Hamiltonian realization. If all periods vanish, fix the identity element $e\in G$ and define

$$H\left(g\right):=-{\int }_{{\gamma }_g}\alpha ,$$

where ${\gamma }_g$ is any smooth path from e to g. The vanishing of the periods guarantees that H is well defined and satisfies $dH=-\alpha$. This completes the proof. □

3. Example: The Symplectic Case

This section illustrates Theorem 1 through explicit computations on the classical symplectic ${\mathrm{Aff}}^{+}\left(1\right)$ Lie group,

3.1. The Affine Group ${\mathrm{Aff}}^{+}\left(1\right)$

In this subsection, we develop in full detail the example of the affine group ${\mathrm{Aff}}^{+}\left(1\right)$, and we interleave, after each key computation, a short interpretation in terms of a 1D ideal fluid with affine symmetries (global scaling and translation).

3.1.1. The Lie Group Structure and Its Fluid Interpretation

Consider the affine group

$${\mathrm{Aff}}^{+}\left(1\right)=\left\{\left(\begin{array}{cc}a& b\\ 0& 1\end{array}\right):a>0,b\in \mathbb{R}\right\},$$

with group law

$$(a,b)(a^{\prime },b^{\prime })=(a{a}^{\prime },b+a{b}^{\prime }).$$

We use global coordinates $(a,b)\in (0,\infty )\times \mathbb{R}$ on ${\mathrm{Aff}}^{+}\left(1\right)$.

• Fluid interpretation. We regard a one-dimensional ideal fluid (for instance, a gas in a 1D tube) whose Lagrangian coordinate x is acted upon by ${\mathrm{Aff}}^{+}\left(1\right)$ via

  $$x\mapsto ax+b.$$

  Here:
  • $a>0$ represents a global scaling of the fluid configuration (homogeneous expansion/compression),
  • $b\in \mathbb{R}$ represents a global translation of the fluid as a whole.

Thus, the pair $(a,b)$ encodes a collective “affine mode’’ of the fluid, describing how the entire 1D medium is stretched and shifted in an idealized, finite-dimensional way.

3.1.2. Left-Invariant Coframe and Symplectic Structure

A left-invariant coframe on ${\mathrm{Aff}}^{+}\left(1\right)$ is given by

$$\alpha ={\displaystyle \frac{da}{a}},\beta ={\displaystyle \frac{db}{a}},$$

which satisfy

$$d\alpha =0,d\beta =-\alpha \wedge \beta .$$

Define the 2-form

$$\omega :=\alpha \wedge \beta ={\displaystyle \frac{1}{a^2}}da\wedge db.$$

Since $d\omega =d(\alpha \wedge \beta )=0$ and $\omega$ is nondegenerate, $({\mathrm{Aff}}^{+}\left(1\right),\omega )$ is a symplectic Lie group.

• Fluid interpretation. The 1-forms $\alpha$ and $\beta$ encode infinitesimal variations of the affine parameters:

  1.

  $\alpha ={\displaystyle \frac{da}{a}}$ measures the relative change of scale of the fluid configuration,

  2.

  $\beta ={\displaystyle \frac{db}{a}}$ measures the effective translation at the current scale.

The symplectic form

$$\omega ={\displaystyle \frac{1}{a^2}}da\wedge db$$

plays the role of an effective phase–space area form for the affine mode: it couples global scaling and translation in a Hamiltonian fashion. From the viewpoint of geometric fluid dynamics, $({\mathrm{Aff}}^{+}\left(1\right),\omega )$ is a finite-dimensional phase space that captures a reduced, affine symmetry mode of a 1D ideal fluid.

3.1.3. A Linear Vector Field and Its Associated 1-Form

Fix $\rho \in \mathbb{R}$ and consider the linear vector field

$$\mathcal{X}=\rho (a-1){\partial }_b.$$

A direct contraction with the symplectic form yields

$${\iota }_{\mathcal{X}}\omega ={\iota }_{\rho (a-1){\partial }_b}\left({\displaystyle \frac{1}{a^2}}da\wedge db\right)={\displaystyle \frac{\rho (a-1)}{a^2}}{\iota }_{{\partial }_b}(da\wedge db).$$

Since

$${\iota }_{{\partial }_b}(da\wedge db)=\left({\iota }_{{\partial }_b}da\right)db-da\left({\iota }_{{\partial }_b}db\right)=0·db-da·1=-da,$$

we obtain

$${\iota }_{\mathcal{X}}\omega =-\rho {\displaystyle \frac{a-1}{a^2}}da=-\rho \left({\displaystyle \frac{1}{a}}-{\displaystyle \frac{1}{a^2}}\right)da.$$

Thus, the associated 1-form is

$${\alpha }_X:={\iota }_{\mathcal{X}}\omega =-\rho \left({\displaystyle \frac{1}{a}}-{\displaystyle \frac{1}{a^2}}\right)da.$$

Fluid interpretation. The vector field $\mathcal{X}=\rho (a-1){\partial }_b$ prescribes a translation velocity of the entire fluid:

• When $a=1$ (no global deformation), the translation speed vanishes;
• When $a>1$ (global expansion), the fluid drifts with speed $\rho (a-1)$;
• When $0<a<1$ (global compression), the drift is reversed.

The 1-form ${\alpha }_X={\iota }_{\mathcal{X}}\omega$ measures how this affine drift interacts with the symplectic structure. In Hamiltonian terms, ${\alpha }_X$ is the candidate to be $-dH$ for some global energy function H. Hence, from the fluid perspective, ${\alpha }_X$ encodes how the affine drift of the 1D fluid mode is compatible with an underlying ideal (energy-preserving) structure.

3.1.4. Symplecticity and Global Hamiltonian Realizability

By Cartan’s identity,

$${\mathcal{L}}_{\mathcal{X}}\omega =d\left({\iota }_{\mathcal{X}}\omega \right)+{\iota }_{\mathcal{X}}d\omega =d{\alpha }_X,$$

since $d\omega =0$. Writing

$${\alpha }_X=f\left(a\right)da,f\left(a\right):=-\rho \left({\displaystyle \frac{1}{a}}-{\displaystyle \frac{1}{a^2}}\right),$$

we have

$$d{\alpha }_X=df\wedge da=f^{\prime }\left(a\right)da\wedge da=0.$$

Hence

$${\mathcal{L}}_{\mathcal{X}}\omega =0,$$

so $\mathcal{X}$ is a symplectic vector field.

Moreover, the underlying manifold $(0,\infty )\times \mathbb{R}$ is simply connected (it is diffeomorphic to ${\mathbb{R}}^2$), so its first de Rham cohomology group vanishes:

$$H_{\mathrm{dR}}^1\left({\mathrm{Aff}}^{+}\left(1\right)\right)=0.$$

Therefore, every closed 1-form is exact. In particular, ${\alpha }_X$ is exact, so there exists a globally defined Hamiltonian function H such that

$${\alpha }_X=-dH.$$

By Theorem 1, the linear symplectic vector field $\mathcal{X}$ is globally Hamiltonian.

• Fluid interpretation. The condition ${\mathcal{L}}_{\mathcal{X}}\omega =0$ means that the affine dynamics preserves the effective phase-space area from $\omega$: the reduced system is conservative in the symplectic sense. For the 1D ideal fluid mode, this says that the combined evolution of scaling a and translation b does not destroy the underlying ideal structure.

The vanishing of $H_{\mathrm{dR}}^1$ implies that there is no topological obstruction to defining a global energy functional H. Hence, every linear symplectic affine mode in this simple setting is automatically generated by a globally defined Hamiltonian $H(a,b)$. This fits the picture of an ideal fluid mode: the reduced dynamics admits a global energy function and is fully Hamiltonian.

3.1.5. Explicit Hamiltonian and Fluid-Mechanical Meaning

To find H, we solve

$${\alpha }_X=-dH,$$

that is,

$$-\rho \left({\displaystyle \frac{1}{a}}-{\displaystyle \frac{1}{a^2}}\right)da=-dH⟺dH=\rho \left({\displaystyle \frac{1}{a}}-{\displaystyle \frac{1}{a^2}}\right)da.$$

Integrating in a gives

$$H(a,b)=\rho \int \left({\displaystyle \frac{1}{a}}-{\displaystyle \frac{1}{a^2}}\right)da=\rho \left(lna+{\displaystyle \frac{1}{a}}\right)+C,$$

where C is a constant. Up to an additive constant, we may take $C=0$ and write

$$H(a,b)=\rho \left(lna+{\displaystyle \frac{1}{a}}\right).$$

A direct check shows

$$dH=\rho \left({\displaystyle \frac{1}{a}}-{\displaystyle \frac{1}{a^2}}\right)da,-dH=-\rho \left({\displaystyle \frac{1}{a}}-{\displaystyle \frac{1}{a^2}}\right)da={\alpha }_X={\iota }_{\mathcal{X}}\omega .$$

Fluid interpretation. The Hamiltonian

$$H\left(a\right)=\rho \left(lna+{\displaystyle \frac{1}{a}}\right)$$

depend only on the global scale a and can be viewed as an effective energy of the affine mode of the 1D ideal fluid:

• The term $lna$ resembles volumetric or compressive energy contributions in continuum mechanics: changing a from 1 to another value has an energetic cost.
• The term $1/a$ penalizes strong compression ($a\ll 1$) and decays as the fluid is expanded ($a\gg 1$).

Thus, $H\left(a\right)$ measures how energetically costly it is to deform the fluid homogeneously by a factor a. The fact that $\mathcal{X}$ is exactly the Hamiltonian vector field of H means that the affine drift $(a,b)\mapsto (a,b(t\left)\right)$ is compatible with an ideal, energy–conserving structure on this reduced phase space.

3.1.6. Hamiltonian Equations and Ideal Fluid Affine Mode

Since $\mathcal{X}=X_H$, the Hamiltonian dynamics generated by H on $({\mathrm{Aff}}^{+}\left(1\right),\omega )$ is

$$\dot{a}=0,\dot{b}=\rho (a-1).$$

For an initial condition $(a\left(0\right),b\left(0\right))=(a_0,b_0)$, the solution is

$$a\left(t\right)=a_0,b\left(t\right)=b_0+t\rho (a_0-1).$$

Hence

$$(a\left(t\right),b\left(t\right))=(a_0,b_0+t\rho (a_0-1)).$$

Fluid interpretation. The reduced affine dynamics of the 1D ideal fluid satisfy:

• The global scale $a_0$ is constant in time: the uniform expansion/compression of the fluid is frozen in this mode.
• The global translation $b\left(t\right)$ evolves linearly with time, with speed $\rho (a_0-1)$. The sign and magnitude of this speed are controlled by the deviation of $a_0$ from the reference scale 1.

The energy $H\left(a_0\right)$ is preserved along trajectories, consistently with the Hamiltonian nature of the model. In this sense, $({\mathrm{Aff}}^{+}\left(1\right),\omega ,H)$ provides a finite-dimensional, fully explicit example of a 1D ideal fluid mode with affine symmetries, where:

• The global deformation (scale a) and drift (translation b) evolve in a way compatible with a symplectic, energy–conserving structure;
• The cohomological criterion of Theorem 1 guarantees the global existence of the affine energy functional H.

4. Hamiltonian Lifting on Odd-Dimensional Lie Groups

Let G be a connected Lie group of odd dimension $2n+1$. Since no symplectic structure can exist globally in this setting, the classical Hamiltonian formalism must be replaced by a different geometric mechanism. In this section, we show that contact geometry provides a canonical and intrinsic framework for Hamiltonian lifting of linear dynamics on odd-dimensional Lie groups [11,13]. As in the symplectic case, linear vector fields generate derivations of the Lie algebra. And, when the group is simply connected, they are in complete correspondence, i.e., the Lie algebra of derivations coincides with the tangent space of the identity element of $Aut\left(\mathfrak{g}\right)$.

Our goal is to characterize when a linear vector field admits a Hamiltonian realization compatible with a contact structure. Next, we use the standard result [18,19] in Proposition 1 to build the Hamiltonian contact vector field. We also explain the role of the Reeb vector field in this construction, as follows.

Proposition 1.

Let $(M^{2n+1},\alpha )$ be a contact manifold with Reeb vector field $R_{\alpha }$. Then, the contact form α determines a canonical splitting

$$TM=\mathbb{R}{R}_{\alpha }\oplus ker\alpha ,$$

where ${\alpha |}_{ker\alpha }$ is a symplectic form on the horizontal distribution $ker\alpha$. For every smooth function $H\in C^{\infty }\left(M\right)$, there exists auniquevector field $X_H\in \mathfrak{X}\left(M\right)$ such that:

$$\begin{array}{ccc}\hfill \alpha \left(X_H\right)& =& H,\hfill \\ \hfill {\iota }_{X_H}d\alpha & =& -dH+\left(R_{\alpha }H\right)\alpha .\hfill \end{array}$$

Equivalently, $X_H$ decomposes uniquely as

$$X_H=H{R}_{\alpha }+Y_H,Y_H\in \mathsf{\Gamma }(ker\alpha ),$$

where the horizontal component $Y_H$ is determined by the symplectic inversion

$${\iota }_{Y_H}\left(d\alpha {|}_{ker\alpha }\right)=-\left(dH\right){|}_{ker\alpha }.$$

Moreover, we have the following assertions:

• The Reeb vector field $R_{\alpha }$ is the distinguished vertical direction of the dynamics;
• The Hamiltonian H prescribes the motion along $R_{\alpha }$;
• The horizontal dynamics is uniquely determined by α on $ker\alpha$, with no gauge ambiguity.

Theorem 2.

Let G be a Lie group of dimension $2n+1$ endowed with a left-invariant contact form α and Reeb vector field $R_{\alpha }$. Let $\mathcal{X}$ be a linear vector field on G satisfying

$${\mathcal{L}}_{\mathcal{X}}\alpha =f\alpha$$

for some smooth function f. Then there exists a unique function $H\in C^{\infty }\left(G\right)$ such that

$$\alpha \left(\mathcal{X}\right)=H$$

and

$${\iota }_{\mathcal{X}}d\alpha =-dH+\left(R_{\alpha }H\right)\alpha .$$

Equivalently, $\mathcal{X}$ coincides with the contact Hamiltonian vector field generated by H.

Proof. 

We prove that a (conformal) contact vector field is automatically a contact Hamiltonian vector field, and we make explicit the role of the Reeb direction. Contact geometry fixes a canonical vertical direction. Since $\alpha$ is a contact form, the Reeb field $R_{\alpha }$ is uniquely characterized by

$$\alpha \left(R_{\alpha }\right)=1,{\iota }_{R_{\alpha }}d\alpha =0.$$

Geometrically, $R_{\alpha }$ is the distinguished direction transverse to the contact hyperplanes $ker\alpha$ and invisible to $d\alpha$. Consequently,

$$TG=\mathbb{R}{R}_{\alpha }\oplus ker\alpha ,$$

and ${d\alpha |}_{ker\alpha }$ is nondegenerate (symplectic) on the horizontal distribution $ker\alpha$.

Next, we consider the Hamiltonian vertical component of the field, the Reeb vector field.

Let $\mathcal{X}$ be the given vector field on G and set

$$H:=\alpha \left(\mathcal{X}\right).$$

Decompose $\mathcal{X}$ uniquely using the canonical splitting:

$$\mathcal{X}=a{R}_{\alpha }+Y,Y\in \mathsf{\Gamma }(ker\alpha ).$$

Applying $\alpha$ and using $\alpha \left(R_{\alpha }\right)=1$ and $\alpha \left(Y\right)=0$, we obtain

$$H=\alpha \left(\mathcal{X}\right)=a,$$

so the vertical coefficient is forced to be $a=H$. Thus

$$\mathcal{X}=H{R}_{\alpha }+Y,Y\in \mathsf{\Gamma }(ker\alpha ).$$

This is the first key geometric point: the Reeb direction provides a canonical vertical component, and the function H prescribes motion along it.

On the other hand, Cartan’s identity determines the horizontal component through

$${\mathcal{L}}_{\mathcal{X}}\alpha ={\iota }_{\mathcal{X}}d\alpha +d\left(\alpha \left(\mathcal{X}\right)\right)={\iota }_{\mathcal{X}}d\alpha +dH.$$

By assumption ${\mathcal{L}}_{\mathcal{X}}\alpha =f\alpha$, hence

$${\iota }_{\mathcal{X}}d\alpha =f\alpha -dH.$$

Now evaluate this identity on the Reeb vector field $R_{\alpha }$. Since $d\alpha (R_{\alpha },·)=0$, the left–hand side vanishes on $R_{\alpha }$:

$$0=\left(f\alpha -dH\right)\left(R_{\alpha }\right)=f\alpha \left(R_{\alpha }\right)-R_{\alpha }\left(H\right)=f-R_{\alpha }\left(H\right).$$

Therefore,

$$f=R_{\alpha }\left(H\right).$$

Substituting back yields the intrinsic contact Hamiltonian identity

$${\iota }_{\mathcal{X}}d\alpha =-dH+\left(R_{\alpha }H\right)\alpha .$$

This is exactly the defining equation for the contact Hamiltonian field with Hamiltonian H.

Finally, the horizontal component Y is uniquely determined because restricting the equation to $ker\alpha$ kills the $\alpha$ term and gives

$${\iota }_Y\left(d\alpha {|}_{ker\alpha }\right)=-\left(dH\right){|}_{ker\alpha }.$$

Since ${d\alpha |}_{ker\alpha }$ is nondegenerate, there is a unique solution $Y\in \mathsf{\Gamma }(ker\alpha )$. Hence, $\mathcal{X}$ coincides with the unique contact Hamiltonian field generated by $H=\alpha \left(\mathcal{X}\right)$, ending the proof. □

Theorem 2 shows that, in odd dimensions, contact geometry provides a canonical Hamiltonian lifting mechanism for linear dynamics. The Reeb vector field serves as a distinguished direction, eliminating the gauge ambiguities that typically arise in presymplectic geometry.

5. Example: The Contact Case

In this section, we illustrate Theorem 2 through an explicit example on the classical Heisenberg Lie group of dimension three. All computations are carried out in left-invariant frames, so that the contact Hamiltonian vector fields can be written in closed form. Each example highlights a different geometric or applied aspect of the theory.

5.1. The Heisenberg Group $H_3$ as a Non-Conservative Fluid Mode

In this subsection, we develop in full detail the contact Hamiltonian example on the classical Heisenberg group $H_3$, interleaving each step of the construction with its interpretation as a reduced fluid mode exhibiting effective non-conservative behavior (entropy or internal energy production).

5.1.1. Configuration Space and Contact Form

We consider the Heisenberg group $H_3\simeq {\mathbb{R}}^3$ with global coordinates $(x,y,z)$ and the 1-form

$$\alpha =dz+\frac{1}{2}(ydx-xdy).$$

Fluid interpretation.

We view $(x,y,z)$ as coordinates on a reduced state space for an effective fluid model, where:

• $(x,y)$ represent horizontal kinematic variables, e.g., two principal modes of motion of a fluid parcel in a reduced 2D plane.
• z represents an internal variable, such as internal energy, entropy, or an accumulated dissipative quantity.

The 1-form $\alpha$ encodes a constitutive relation between $(x,y)$ and z: it specifies how infinitesimal horizontal changes in $(x,y)$ are coupled to changes in the internal variable z.

5.1.2. Contact Structure

We first check that $\alpha$ is a contact form. Computing the exterior derivative:

$$d\alpha =d\left(dz+\frac{1}{2}(ydx-xdy)\right)=\frac{1}{2}\left(dy\wedge dx-dx\wedge dy\right)=dx\wedge dy.$$

Then

$$\alpha \wedge d\alpha =\left(dz+\frac{1}{2}(ydx-xdy)\right)\wedge dx\wedge dy=dz\wedge dx\wedge dy+\frac{1}{2}(ydx-xdy)\wedge dx\wedge dy.$$

Since

$$(ydx-xdy)\wedge dx\wedge dy=ydx\wedge dx\wedge dy-xdy\wedge dx\wedge dy=0,$$

we obtain

$$\alpha \wedge d\alpha =dz\wedge dx\wedge dy,$$

which is nowhere vanishing. Thus, $\alpha$ is a contact form on $H_3$ and $(H_3,\alpha )$ is a contact manifold.

• Fluid interpretation.

The non-vanishing of $\alpha \wedge d\alpha$ means that the hyperplanes $ker\alpha$ define a maximally non-integrable distribution: there is no foliation by surfaces tangent to $ker\alpha$. In fluid terms, this reflects a non-holonomic constraint or a built-in relation between horizontal motion and internal evolution. The manifold $(H_3,\alpha )$ therefore provides a natural geometric stage for modeling fluids where mechanical and internal variables are intrinsically coupled.

5.1.3. Reeb Field

The Reeb vector field $R_{\alpha }$ is uniquely determined by

$$\alpha \left(R_{\alpha }\right)=1,{\iota }_{R_{\alpha }}d\alpha =0.$$

We have

$$d\alpha =dx\wedge dy,$$

so for any vector field $V=V_x{\partial }_x+V_y{\partial }_y+V_z{\partial }_z$,

$${\iota }_{V}d\alpha ={\iota }_V(dx\wedge dy)=V_x{\iota }_{{\partial }_x}(dx\wedge dy)+V_y{\iota }_{{\partial }_y}(dx\wedge dy)+V_z{\iota }_{{\partial }_z}(dx\wedge dy).$$

Using

$${\iota }_{{\partial }_x}(dx\wedge dy)=dy,{\iota }_{{\partial }_y}(dx\wedge dy)=-dx,{\iota }_{{\partial }_z}(dx\wedge dy)=0,$$

we obtain

$${\iota }_{V}d\alpha =V_{x}dy-V_{y}dx.$$

Hence, ${\iota }_{V}d\alpha =0$ if and only if $V_x=V_y=0$, i.e., $V=V_z{\partial }_z$. Imposing $\alpha \left(V\right)=1$ gives

$$\alpha \left({\partial }_z\right)=dz\left({\partial }_z\right)+\frac{1}{2}\left(ydx\left({\partial }_z\right)-xdy\left({\partial }_z\right)\right)=1+0=1.$$

Therefore

$$R_{\alpha }={\partial }_z.$$

Fluid interpretation.

The Reeb field $R_{\alpha }={\partial }_z$ is the distinguished internal direction: it changes only the internal variable z, leaving the horizontal coordinates $(x,y)$ fixed. Since $R_{\alpha }$ is invisible to $d\alpha$ (it lies in the kernel of $d\alpha$), it does not affect the instantaneous horizontal symplectic structure. In a fluid setting, $R_{\alpha }$ represents the direction along which entropy or internal energy can accumulate without directly modifying the horizontal mechanics.

5.1.4. Horizontal Distribution and Adapted Frame

The contact distribution is

$$ker\alpha =\{V\in T{H}_3:\alpha \left(V\right)=0\}.$$

Define

$$E_2:={\partial }_x-{\displaystyle \frac{y}{2}}{\partial }_z,E_3:={\partial }_y+{\displaystyle \frac{x}{2}}{\partial }_z.$$

We check that $E_2,E_3\in ker\alpha$:

$$\alpha \left(E_2\right)=dz\left(E_2\right)+\frac{1}{2}\left(ydx\left(E_2\right)-xdy\left(E_2\right)\right)=\left(-{\displaystyle \frac{y}{2}}\right)+\frac{1}{2}\left(y·1-x·0\right)=0,$$

$$\alpha \left(E_3\right)=dz\left(E_3\right)+\frac{1}{2}\left(ydx\left(E_3\right)-xdy\left(E_3\right)\right)=\left({\displaystyle \frac{x}{2}}\right)+\frac{1}{2}\left(y·0-x·1\right)=0.$$

Moreover, using $d\alpha =dx\wedge dy$ and the fact that $E_2$ and $E_3$ differ from ${\partial }_x,{\partial }_y$ only by ${\partial }_z$–components (which do not contribute to $dx\wedge dy$), we have

$$d\alpha (E_2,E_3)=dx\left(E_2\right)dy\left(E_3\right)-dx\left(E_3\right)dy\left(E_2\right)=\left(1\right)\left(1\right)-\left(0\right)\left(0\right)=1.$$

Thus, $ker\alpha =\mathrm{span}\{E_2,E_3\}$ and ${d\alpha |}_{ker\alpha }$ is nondegenerate (symplectic). The tangent bundle splits as

$$T{H}_3=\mathbb{R}{R}_{\alpha }\oplus ker\alpha =\mathrm{span}\left\{{\partial }_z\right\}\oplus \mathrm{span}\{E_2,E_3\}.$$

Fluid interpretation.

The splitting

$$T{H}_3=\mathbb{R}{R}_{\alpha }\oplus ker\alpha$$

separates:

• the horizontal mechanical space $ker\alpha$, where $d\alpha$ is symplectic and encodes ideal, conservative dynamics in $(x,y)$,
• from the vertical internal direction $\mathbb{R}{R}_{\alpha }$, where internal quantities evolve.

The frame $(E_2,E_3)$ spans the horizontal (mechanical) modes of the fluid, while $R_{\alpha }$ captures the internal (thermodynamic) mode.

5.1.5. Contact Hamiltonian Field

Let

$$H(x,y,z)={\displaystyle \frac{1}{2}}(x^2+y^2).$$

By the general contact Hamiltonian theory, the vector field $X_H$ associated to H is uniquely determined by

$$\left\{\begin{array}{c}\alpha \left(X_H\right)=H,\hfill \\ {\iota }_{X_H}d\alpha =-dH+\left(R_{\alpha }H\right)\alpha .\hfill \end{array}\right.$$

(1)

We decompose

$$X_H=H{R}_{\alpha }+Y_H,Y_H\in \mathsf{\Gamma }(ker\alpha ).$$

Since $R_{\alpha }={\partial }_z$ and H does not depend on z,

$$R_{\alpha }H={\partial }_{z}H=0.$$

Then, the second equation in (1) reduces to

$${\iota }_{X_H}d\alpha ={\iota }_{H{R}_{\alpha }+Y_H}d\alpha ={\iota }_{Y_H}d\alpha =-dH.$$

We have

$$dH=xdx+ydy,\Rightarrow -dH=-xdx-ydy.$$

Write $Y_H$ in the horizontal frame:

$$Y_H=a{E}_2+b{E}_3.$$

Then

$${\iota }_{Y_H}d\alpha ={\iota }_{a{E}_2+b{E}_3}(dx\wedge dy)=a{\iota }_{E_2}(dx\wedge dy)+b{\iota }_{E_3}(dx\wedge dy).$$

Using again that the ${\partial }_z$-components do not contribute,

$${\iota }_{E_2}(dx\wedge dy)={\iota }_{{\partial }_x}(dx\wedge dy)=dy,{\iota }_{E_3}(dx\wedge dy)={\iota }_{{\partial }_y}(dx\wedge dy)=-dx,$$

so

$${\iota }_{Y_H}d\alpha =ady-bdx.$$

Equating this with $-dH=-xdx-ydy$ yields

$$ady-bdx=-xdx-ydy,$$

and hence

$$-b=-x\Rightarrow b=x,a=-y.$$

Therefore

$$Y_H=-y{E}_2+x{E}_3.$$

Now substitute $E_2,E_3$:

$$-y{E}_2+x{E}_3=-y\left({\partial }_x-{\displaystyle \frac{y}{2}}{\partial }_z\right)+x\left({\partial }_y+{\displaystyle \frac{x}{2}}{\partial }_z\right)=-y{\partial }_x+x{\partial }_y+\left(\frac{y^2}{2}+\frac{x^2}{2}\right){\partial }_z.$$

Adding the Reeb component $H{R}_{\alpha }={\displaystyle \frac{1}{2}}(x^2+y^2){\partial }_z$, we obtain

$$\begin{array}{cc}\hfill X_H& =H{R}_{\alpha }+Y_H\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}(x^2+y^2){\partial }_z-y{\partial }_x+x{\partial }_y+\left(\frac{y^2}{2}+\frac{x^2}{2}\right){\partial }_z\hfill \\ \hfill & =-y{\partial }_x+x{\partial }_y+(x^2+y^2){\partial }_z.\hfill \end{array}$$

Thus

$$X_H=-y{\partial }_x+x{\partial }_y+(x^2+y^2){\partial }_z.$$

Fluid interpretation. The horizontal component

$$Y_H=-y{E}_2+x{E}_3$$

generates a planar rotation in $(x,y)$, while the vertical Reeb component

$$H{R}_{\alpha }=\frac{1}{2}(x^2+y^2){\partial }_z$$

drives the internal variable z at a rate proportional to the horizontal “kinetic energy” $x^2+y^2$. Thus, the same Hamiltonian H that governs the ideal planar motion also controls the rate at which internal energy or entropy is produced along the Reeb direction.

Equations of motion and solution. The contact Hamiltonian system $\dot{q}=X_H\left(q\right)$ is

$$\dot{x}=-y,\dot{y}=x,\dot{z}=x^2+y^2.$$

Differentiating the first two equations,

$$\ddot{x}=-\dot{y}=-x,\ddot{y}=\dot{x}=-y,$$

so

$$x\left(t\right)=rcos(t+\phi ),y\left(t\right)=rsin(t+\phi ),$$

for some constants $r\ge 0$ and $\phi \in \mathbb{R}$ determined by the initial condition $(x_0,y_0)$:

$$r=\sqrt{x_0^2+y_0^2},\phi =arg(x_0+i{y}_0).$$

In particular,

$$x^2\left(t\right)+y^2\left(t\right)=r^2=\mathrm{constant}.$$

Then the equation for z becomes

$$\dot{z}=x^2+y^2=r^2,$$

so

$$z\left(t\right)=z_0+r^{2}t.$$

Fluid interpretation. The detailed computation above shows that:

• The horizontal variables $(x,y)$, lying in $ker\alpha$, follow a uniform rotation with

  $$x^2\left(t\right)+y^2\left(t\right)=r^2=\mathrm{constant},$$

  which may be interpreted as a conserved reduced kinetic energy of a planar fluid mode.
• The vertical variable z evolves along the Reeb direction according to

  $$\dot{z}=x^2+y^2=r^2,z\left(t\right)=z_0+r^{2}t,$$

  and monotonically accumulates a quantity proportional to the horizontal kinetic energy.

This makes $(H_3,\alpha ,H)$ a prototypical example of an effective non-conservative fluid mode: the horizontal (mechanical) motion is ideal and energy-preserving on $ker\alpha$, while the internal direction z registers an irreversible growth of a thermodynamic-type quantity (e.g., entropy or internal energy). The contact Hamiltonian structure encodes this behavior geometrically, and the Reeb field $R_{\alpha }={\partial }_z$ plays the role of a distinguished internal direction along which dissipation or entropy production is recorded.

6. Conclusions

In this paper, we have developed a global theory of linear Hamiltonian vector fields on Lie groups endowed with Hamiltonian-type geometric structures, with a particular emphasis on their interpretation as reduced models of fluid motion.

In the even-dimensional, symplectic setting, Theorem 1 provides a sharp and intrinsic characterization of when a linear symplectic vector field on a symplectic Lie group $(G,\omega )$ is globally Hamiltonian.

The criterion reduces the problem to a single cohomological obstruction, namely the de Rham class $\left[{\iota }_{\mathcal{X}}\omega \right]\in H_{\mathrm{dR}}^1\left(G\right)$, and shows that on simply connected groups, this obstruction vanishes automatically.

From the viewpoint of geometric fluid dynamics, this result ensures that linear dynamics preserving a symplectic structure-and hence an ideal energy-momentum form–admit a globally defined energy functional on a wide class of symmetry groups.

The explicit affine example ${\mathrm{Aff}}^{+}\left(1\right)$ illustrates this philosophy by realizing a one-dimensional ideal fluid mode with affine symmetries, where the global scale and translation form a symplectic phase space and the Hamiltonian depends only on the homogeneous deformation parameter.

In the odd-dimensional case, where no global symplectic structure exists, we have shown that contact geometry provides the natural replacement for Hamiltonian lifting.

Theorem 2 establishes that, on a contact Lie group $(G,\alpha )$, any linear vector field satisfying a conformal contact condition ${\mathcal{L}}_{\mathcal{X}}\alpha =f\alpha$ is necessarily a contact Hamiltonian vector field for a uniquely determined Hamiltonian $H=\alpha \left(\mathcal{X}\right)$.

The Reeb vector field $R_{\alpha }$ furnishes a canonical vertical direction, resolving the gauge ambiguities typical of degenerate or presymplectic systems. Physically, this vertical direction can be interpreted as an internal or entropic coordinate along which non-conservative effects occur.

The Heisenberg example $(H_3,\alpha ,H)$ makes this contact framework completely explicit. The horizontal variables $(x,y)$ evolve as an ideal planar mode with conserved quadratic energy, while the internal variable z (the Reeb coordinate) accumulates this energy linearly in time.

This yields a finite-dimensional contact Hamiltonian model of an effective non-conservative fluid mode: conservative horizontal mechanics coupled to an irreversible internal evolution, fully encoded by the contact structure and its Reeb field.

Taken together, the symplectic and contact theories developed here provide a unified geometric framework for linear Hamiltonian dynamics on Lie groups that naturally accommodates both ideal (energy-conserving) and effective (non-conservative) fluid behaviors.

7. Perspectives and Future Work

The results presented here suggest several directions for further research, both from the standpoint of geometric analysis and from the perspective of fluid dynamics and control.

On the symplectic side, it would be natural to embed the finite-dimensional models studied here into broader Lie-Poisson and coadjoint-orbit formulations of ideal fluids and continua. In particular:

• Relating the cohomological obstruction $\left[{\iota }_{\mathcal{X}}\omega \right]$ in Theorem 1 to invariants in the dual of the Lie algebra ${\mathfrak{g}}^{*}$, and to conserved quantities in Euler–Poincaré or Euler–Arnold equations would clarify the precise status of linear modes such as the affine ideal fluid example within the hierarchy of reduced fluid models.
• Extending the explicit analysis to other symplectic Lie groups (for instance, higher-dimensional affine groups, semidirect products like $SE\left(2\right){\times }_s\mathbb{R}$, or symplectic subgroups of diffeomorphism groups) could yield new families of ideal fluid modes with clear geometric structure and explicitly computable Hamiltonians.

On the contact side, the Heisenberg model highlights how the energycontact Hamiltonian dynamics can encode non-conservative effects such as entropy or internal energy production. A number of extensions suggest themselves:

• Generalizing to higher-dimensional Heisenberg-type groups $H_{2n+1}$ and other contact Lie groups (e.g., contact structures on $SE\left(2\right)$ or $SL(2,\mathbb{R})$) could provide a family of finite-dimensional prototypes for multi-mode effective fluid models, where several horizontal modes jointly feed one or more internal variables.
• Connecting the contact lifting theorem to recent developments in geometric formulations of dissipative and thermodynamic systems (e.g., Lagrange-Dirac, or generic frameworks) may help to systematically derive reduced fluid models with built-in dissipation from more detailed continuum descriptions.

Overall, we expect that the combination of the global linear theory developed here with more elaborate reduction, control, and contact–thermodynamic techniques will lead to a rich class of structured, finite-dimensional models for complex fluid phenomena, firmly rooted in the geometry of Lie groups and their Hamiltonian and contact dynamics.