Jacobi-Lie systems: Fundamentals and low-dimensional classification

A Lie system is a system of differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields, a Vessiot-Guldberg Lie algebra. We define and analyze Lie systems possessing a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields relative to a Jacobi manifold, the hereafter called Jacobi-Lie systems. We classify Jacobi-Lie systems on $\mathbb{R}$ and $\mathbb{R}^2$. Our results shall be illustrated through examples of physical and mathematical interest.


Introduction
The inspection of Lie systems traces back to the 19th century when Lie laid down the fundamentals on the study of systems of first-order ordinary differential equations admitting superposition rules, i.e., functions allowing us to describe their general solutions in terms of a finite generic family of particular solutions and some parameters [18].
Subsequently, Lie systems were hardly ever investigated for almost a century until Winternitz retook their study [6,23]. Since then, many authors have been investigating them [1,3,5,7,11]. Although Lie systems rarely occur in the physics and mathematics literature, they appear in relevant problems and enjoy interesting geometric properties, which motivate their study [6,7,8,19]. Some attention has been paid to Lie systems admitting a Vessiot-Guldberg Lie algebra (VG Lie algebra) of Hamiltonian vector fields with respect to certain geometric structure [1,3,4,5,7,12,19]. The first step was to study Lie systems with VG Lie algebras of Hamiltonian vector fields with respect to a symplectic form [5]. Lie systems admitting a VG Lie algebra of Hamiltonian vector fields with respect to a Poisson bivector were dubbed as Lie-Hamilton systems [7]. Next, Lie systems possessing VG Lie algebras of Hamiltonian vector fields relative to Dirac structures [4] and ksymplectic structures [19] were investigated. All these different geometries allow one to obtain superposition rules, constants of motion and Lie symmetries for Lie systems by means of algebraic and geometric methods, e.g., the celebrated superposition rule for Riccati equations can be obtained through the Casimir element of sl(2, R) [3].
Following the above research line, we here study Lie systems with VG Lie algebras of Hamiltonian vector fields with respect to Jacobi manifolds. Roughly speaking, a Jacobi manifold is a manifold N endowed with a local Lie algebra (C ∞ (N), {·, ·}) [9,10,14,15,20]. Using that Poisson manifolds are a particular case of Jacobi manifolds, we can consider Jacobi-Lie systems as a generalization of Lie-Hamilton systems. Although each Jacobi manifold gives rise to an associated Dirac manifold, not all Hamiltonian vector fields with respect to the Jacobi manifold become Hamiltonian with respect to its associated Dirac manifold (cf. [9]). Hence, not every Jacobi-Lie system can be straightforwardly understood as a Dirac-Lie system. Even in that case, the Jacobi manifold allows us to construct a Dirac manifold to study the system. The main difference between Jacobi-Lie systems and Lie-Hamilton systems is that Jacobi manifolds do not naturally give rise to Poisson brackets on a space of smooth functions on the manifold, which makes difficult to prove analogues and/or extensions of the results for Lie-Hamilton systems.
In this work, we extend to Jacobi-Lie systems some of the main structures found for Lie-Hamilton systems, e.g., Lie-Hamiltonians [7], and we classify all Jacobi-Lie systems on R and R 2 by determining all VG Lie algebras of Hamiltonian vector fields with respect to Jacobi manifolds on R and R 2 . This is achieved by using the local classification of Lie algebras of vector fields on R and R 2 derived by Lie [17] and improved by González-López, Kamran and Olver (GKO) [13] (see also [2]). Summarizing, we obtain that every Lie system on the real line is a Jacobi-Lie system and we show that every Jacobi-Lie system on the plane admits a VG Lie algebra diffeomorphic to one of the 14 classes indicated in Table 1 below.

Fundamentals on Lie-Hamilton systems
All structures throughout this work are assumed to be smooth, real and globally defined. Let V be a vector space and [·, ·] : V × V → V a Lie bracket. We denote by (V, [·, ·]) the corresponding real Lie algebra. Given subsets A 1 , A 2 ⊂ V , we write [A 1 , A 2 ] for the real vector space spanned by the Lie brackets between elements of A 1 and A 2 , respectively. We define Lie(A 1 , V, [·, ·]) to be the smallest Lie subalgebra of V containing A 1 . To abbreviate, we use Lie(A 1 ) and V to represent Lie(A 1 , V, [·, ·]) and (V, [·, ·]), correspondingly.
. The minimal Lie algebra of X is the smallest real Lie algebra, V X , containing {X t } t∈R , namely V X = Lie({X t } t∈R ). An integral curve of X is an integral curve γ : R → R × N of the suspension of X, i.e. the vector field ∂/∂t + X(t, x) on R × N. For every γ of the form t → (t, x(t)), we have that dx dt (t) = (X • γ)(t).
This system is referred to as the associated system of X. Conversely, every system of first-order differential equations in the normal form describes the integral curves (t, x(t)) of a unique t-dependent vector field X. So, we can use X to denote both the t-dependent vector field and the associated system. Definition 2.1. A Lie system is a system X whose V X is finite-dimensional [6].
Note that if X admits a VG Lie algebra, V X is finite-dimensional and conversely.
Example 2.2. Consider the following system of Riccati equations [6] dx with a 0 (t), a 1 (t), a 2 (t) being arbitrary t-dependent functions. System (1) is associated to the t-dependent vector field X R = a 0 (t)X 1 + a 1 (t)X 2 + a 2 (t)X 3 , where Hence, X R takes values in a VG Lie algebra X 1 , X 2 , X 3 ≃ sl(2, R). Lie proved that each Lie system on the real line is, up to a local change of variables, a particular case of system (1) for n = 1 [6,17].
Let Γ(Λ 2 T N) be the space of sections of Λ 2 T N. A Poisson manifold is a pair (N, Λ), with Λ being a bivector field on N, i.e., Λ ∈ Γ(Λ 2 T N), satisfying that [Λ, Λ] SN = 0, where [·, ·] SN is the so-called Schouten-Nijenhius bracket [21]. The bivector Λ, the referred to as Poisson bivector, induces a bundle morphism Λ : α x ∈ T * N → Λ x (α x , ·) ∈ T N. We say that a vector field X on N is a Hamiltonian vector field with respect to (N, Λ) if it can be brought into the form X = Λ(df ) for a function f ∈ C ∞ (N). We call f a Hamiltonian function for X. Conversely, every function f is the Hamiltonian function of a unique vector field X f , its Hamiltonian vector field. This gives rise to the bracket {·, ·} Λ : . It can be proved that every Poisson bracket on C ∞ (N) amounts to a Poisson bivector on N [21]. Definition 2.3. A Lie-Hamilton system is a Lie system X whose V X consists of Hamiltonian vector fields with respect to a Poisson bivector [7]. Example 2.4. Let us reconsider the Lie system (1) for n = 4 and Hence, the vector fields X 1 , X 2 , X 3 given in (2) are Hamiltonian with respect to (O, Λ R ) and X R becomes a Lie-Hamilton system. The relevance of this result is due to the fact that it allows one the obtain the superposition rule for Riccati equations through a Casimir of sl(2, R) [3].

Jacobi manifolds
Jacobi manifolds where independently introduced by Kirillov and Lichnerowicz [14,16]. We now briefly survey their most fundamental properties. Several known results will be illustrated during the proof of the main results of the paper.
Definition 3.1. A Jacobi manifold is a triple (N, Λ, R), where Λ is a bivector field on N and R is a vector field, the referred to as Reeb vector field, satisfying Example 3.2. Every Poisson manifold (N, Λ) can be considered as a Jacobi manifold (N, Λ, R = 0).
The continuous Heisenberg group [22] can be described as the space of matrices endowed with the standard matrix multiplication, where {x, y, z} is the natural coordinate system on H induced by (3). Consider the bivector field on H given by and the vector field R H ≡ ∂/∂z. After a simple calculation, we obtain that Definition 3.4. We say that X is a Hamiltonian vector field with respect to the Jacobi manifold (N, Λ, R) if there exists a function f ∈ C ∞ (N) such that In this case, f is said to be a Hamiltonian function of X and we write X = X f . If f is also a first-integral of R, we say that f is a good Hamiltonian function and we call X f a good Hamiltonian vector field.
Example 3.5. Given the Jacobi manifold (H, Λ H , R H ) and the vector field X L 1 ≡ ∂/∂x, we have that 1 is a Hamiltonian vector field with Hamiltonian function h L 1 = −y with respect to (H, Λ H , R H ).
Each function gives rise to a unique Hamiltonian vector field. Nevertheless, each vector field may admit several Hamiltonian functions. This last result will be illustrated afterwards within relevant results concerning Jacobi-Lie systems. We write Ham(Λ, R) for the space of Hamiltonian vector fields relative to (N, Λ, R). It can be proved that Ham(Λ, R) is a Lie algebra with respect to the standard Lie bracket of vector fields. Additionally, a Jacobi manifold allows us to define a Lie bracket on C ∞ (N) given by {f, g} Λ,R = Λ(df, dg) + f Rg − gRf.

This Lie bracket becomes a Poisson bracket if and only if
is a Lie algebra morphism. It is important to emphasize that it may not be injective.

Jacobi-Lie systems
We now introduce Jacobi-Lie systems as Lie systems admitting a VG Lie algebra of Hamiltonian vector fields relative to a Jacobi manifold.
Consider now the system on H given by for arbitrary t-dependent functions b 1 (t), b 2 (t) and b 3 (t). Since the associated tdependent vector field takes values in a finitedimensional Lie algebra of vector fields, the system X H is a Lie system. The interest of X H is due to its appearance in the solution of the so-called quantum Lie systems as well as Lie systems admitting a VG Lie algebra isomorphic to h (cf. [6]).
Let us show that system (5) gives rise to a Jacobi-Lie systems. Consider the bivector Λ H and the vector field R H on H given in Example 3.3. Then, That is, X L 1 , X L 2 and X L 3 are Hamiltonian vector fields with Hamiltonian functions h 1 = −y, h 2 = x and h 3 = 1, respectively. Hence, we obtain that (H, Λ H , R H , X H ) is a Jacobi-Lie system. It is remarkable that each Hamiltonian function h i is a firstintegral of X L i and R H for i = 1, 2, 3, respectively. Proof. First, we prove that the Jacobi bracket of two good Hamiltonian functions is a good Hamiltonian function. For general functions u 1 , u 2 ∈ C ∞ (N), we have If u 1 , u 2 ∈ G(N, Λ, R), then Ru 1 = Ru 2 = 0. Using this and [Λ, R] SN = 0, we obtain Hence, {u 1 , u 2 } Λ,R ∈ G(N, Λ, R), which becomes a Lie algebra relative to {·, ·} Λ,R . Note also that R(u 1 · u 2 ) = 0 and u 1 · u 2 ∈ G(N, Λ, R).
Given an arbitrary u 1 ∈ G(N, Λ, R) and any u 2 , u 3 ∈ C ∞ (N), we have that is a finite-dimensional real Lie algebra. Given a system X on N, we say that X admits a Jacobi-Lie Hamiltonian (N, Λ, R, h) if X t is a Hamiltonian vector field with Hamiltonian function h t (with respect to (N, Λ, R)) for each t ∈ R.
is a Hamiltonian function for each X H t in (5), with t ∈ R. In addition, In other words, the functions {h t } t∈R span a finite-dimensional real Lie algebra of functions with respect to the Poisson bracket (4) induced by the Jacobi manifold. Thus, X H admits a Jacobi-Lie Hamiltonian (N, Λ H , R H , h).
Theorem 5.3. Given a Jacobi-Lie Hamiltonian (N, Λ, R, h), the system X of the form X t = X ht , ∀t ∈ R, is a Jacobi-Lie system. If X is a Lie system whose {X t } t∈R are good Hamiltonian vector fields, then it admits a Jacobi-Lie Hamiltonian.
Proof. Let us prove the direct part. By assumption, the Hamiltonian functions ) is a finite-dimensional Lie algebra. Hence, X takes values in a finite-dimensional Lie algebra of Hamiltonian vector fields and it is a Jacobi-Lie system.
Let us prove the converse. Since the elements of {X t } t∈R are good Hamiltonian vector fields by assumption and Lie({X t } t∈R ) = V X , every element of V X is a good Hamiltonian vector field and we can choose a basis X 1 , . . . , X r of V X with good Hamiltonian functions h 1 , . . . , h r . The Jacobi bracket Since [X i , X j ] = r k=1 c ijk X k for certain constants c ijk , we obtain that each function is the difference of two good Hamiltonian functions with the same Hamiltonian vector field. Hence, {s ij , h} Λ,R = 0 for all h ∈ C ∞ (N). Using this, we obtain that the linear space generated by h 1 , . . . , h r , s ij , with 1 ≤ i < j ≤ r, is a finite-dimensional Lie algebra relative to {·, ·} Λ,R . If X = r α=1 b α (t)X α , then (N, Λ, R, h = r α=1 b α (t)h α ) is a Jacobi-Lie Hamiltonian for X.
One of the relevant properties of the Jacobi-Lie Hamiltonians is given by the following proposition, whose proof is straightforward. 6 Jacobi-Lie systems on low dimensional manifolds Let us prove that every Lie system on the real line gives rise to a Jacobi-Lie system. In the case of two-dimensional manifolds, we display, with the aid of the GKO classification [2,13] of Lie algebras of vector fields on the plane given in Table 1, all the possible VG Lie algebras related to Jacobi-Lie systems on the plane.
Let us show that (1) can be associated with a Jacobi-Lie system for n = 1, which proves that every Lie system on the real line can be considered as a Jacobi-Lie system. Recall that (1) is a Lie system with a VG Lie algebra V spanned by (2). Observe that V consists of Hamiltonian vector fields with respect to a Jacobi manifold on R given by Λ = 0 and R = ∂ ∂x 1 . Indeed, the vector fields X 1 , X 2 , X 3 ∈ V admit Hamiltonian functions We now classify Jacobi-Lie systems (R 2 , Λ, R, X), where we may assume Λ and R to be locally equal or different from zero. There exists just one Jacobi-Lie system with Λ = 0 and R = 0: (R 2 , Λ = 0, R = 0, X = 0). Jacobi-Lie systems of the form (R 2 , Λ = 0, R = 0) are Lie-Hamilton systems, whose VG Guldberg Lie algebras were obtained in [2]. In Table 1 we indicate these cases by writing Poisson. A Jacobi-Lie system (R 2 , Λ = 0, R = 0, X) is such that if Y ∈ V X , then Y = f R for certain f ∈ C ∞ (R 2 ). All cases of this type can be easily obtained out of the bases given in Table 1. We describe them by writing (0, R) at the last column. Propositions 2 and 3 below, show that the VG Lie algebras of Table 1 that do not fall into the mentioned categories are not VG Lie algebras of Hamiltonian vector fields with respect to Jacobi manifolds (R 2 , Λ = 0, R = 0). This means that every (R 2 , Λ, R, X) admits a VG Lie algebra belonging to one of the previously mentioned classes 2 . Lemma 6.1. Every Jacobi manifold on the plane with R = 0 and Λ = 0 admits a local coordinate system {s, t} where R = ∂ s and Λ = ∂ s ∧ ∂ t .
Definition 6.2. We call the local coordinate variables {s, t} of the above lemma local rectifying coordinates of the Jacobi manifold on the plane. Lemma 6.3. Let (R 2 , Λ, R) be a Jacobi manifold with R ξ = 0 and Λ ξ = 0 at every ξ ∈ R 2 . The Lie algebra morphism φ : C ∞ (R 2 ) → Ham(R 2 , Λ, R) has non-trivial kernel. On local rectifying coordinates, we have ker φ = e t .
Proof. If f ∈ C ∞ (R 2 ) belongs to ker φ, then Λ(df ) + f R = 0. In local rectifying coordinates, we get Proposition 2. If a Lie system on the plane is related to a VG Lie algebra V of vector fields containing two non-zero vector fields X 1 , X 2 satisfying [X 1 , X 2 ] = X 1 and X 1 ∧ X 2 = 0, then V does not consist of Hamiltonian vector fields relative to any Jacobi manifold with R = 0 and Λ = 0.
Proof. Let us take a rectifying coordinate system for (R 2 , Λ, R). Since φ is a morphism of Lie algebras, we get that X 1 , X 2 amount to the existence of non-zero functions h 1 and h 2 such that {h 1 , h 2 } Λ,R = h 1 + g, where g ∈ ker φ. So, Meanwhile, X 1 ∧ X 2 = 0 implies that Using local rectifying coordinates, we see that Λ(dh i ) = (Rh i )∂ t − ∂ t h i R and R ∧ Λ(dh i ) = (Rh i )Λ for i = 1, 2. Hence, This amounts to {h 1 , h 2 } Λ,R = 0, which implies that 0 = h 1 + g and X 1 = 0. This is impossible by assumption and X 1 and X 2 cannot be Hamiltonian.
This implies that dµ 1 ∧ dµ 2 = 0. Since µ 1 is a first-integral for X 1 and µ 2 is a firstintegral for X 2 , this means that X 1 ∧ X 2 = 0, which is impossible by assumption. This finishes the proof. Table 1: Functions 1, ξ 1 (x), . . . , ξ r (x) are linearly independent and η 1 (x), . . . , η r (x) form a basis of solutions for a system of r ≥ 1 linear differential equations in normal form with constant coefficients. Notice that g 1 ⋉ g 2 stands for the semi-direct sum of g 1 by g 2 , i.e. g 2 is an ideal of g 1 ⋉ g 2 .