Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems

Some positive answers to the problem of endowing a dynamical system with a Hamiltonian formulation are presented within the class of Poisson structures in a geometric framework. We address this problem on orientable manifolds and by using decomposable Poisson structures. In the first case, the exist...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2022
Main Authors:	Avendaño-Camacho, Misael, García-Mendoza, Claudio César, Ruíz-Pantaleón, José Crispín, Velasco-Barreras, Eduardo
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Language:	English
Published:	Інститут математики НАН України 2022
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/211630
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Cite this:	Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems. Misael Avendaño-Camacho, Claudio César García-Mendoza, José Crispín Ruíz-Pantaleón and Eduardo Velasco-Barreras. SIGMA 18 (2022), 038, 29 pages

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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author	Avendaño-Camacho, Misael García-Mendoza, Claudio César Ruíz-Pantaleón, José Crispín Velasco-Barreras, Eduardo
author_facet	Avendaño-Camacho, Misael García-Mendoza, Claudio César Ruíz-Pantaleón, José Crispín Velasco-Barreras, Eduardo
citation_txt	Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems. Misael Avendaño-Camacho, Claudio César García-Mendoza, José Crispín Ruíz-Pantaleón and Eduardo Velasco-Barreras. SIGMA 18 (2022), 038, 29 pages
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	Some positive answers to the problem of endowing a dynamical system with a Hamiltonian formulation are presented within the class of Poisson structures in a geometric framework. We address this problem on orientable manifolds and by using decomposable Poisson structures. In the first case, the existence of a Hamiltonian formulation is ensured under the vanishing of some topological obstructions, improving a result of Gao. In the second case, we apply a variant of the Hojman construction to solve the problem for vector fields admitting a transversally invariant metric and, in particular, for infinitesimal generators of proper actions. Finally, we also consider the hamiltonization problem for Lie group actions and give solutions in the particular case in which the acting Lie group is a low-dimensional torus.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 038, 29 pages Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems Misael AVENDAÑO-CAMACHO, Claudio César GARCÍA-MENDOZA, José Crisṕın RUÍZ-PANTALEÓN and Eduardo VELASCO-BARRERAS Departamento de Matemáticas, Universidad de Sonora, México E-mail: misael.avendano@unison.mx, a214200511@unison.mx, jose.ruiz@unison.mx, eduardo.velasco@unison.mx Received March 02, 2021, in final form May 10, 2022; Published online May 20, 2022 https://doi.org/10.3842/SIGMA.2022.038 Abstract. Some positive answers to the problem of endowing a dynamical system with a Hamiltonian formulation are presented within the class of Poisson structures in a geomet- ric framework. We address this problem on orientable manifolds and by using decomposable Poisson structures. In the first case, the existence of a Hamiltonian formulation is ensured under the vanishing of some topological obstructions, improving a result of Gao. In the sec- ond case, we apply a variant of the Hojman construction to solve the problem for vector fields admitting a transversally invariant metric and, in particular, for infinitesimal generators of proper actions. Finally, we also consider the hamiltonization problem for Lie group actions and give solutions in the particular case in which the acting Lie group is a low-dimensional torus. Key words: Hamiltonian formulation; Poisson manifold; first integral; unimodularity; trans- versally invariant metric; symmetry 2020 Mathematics Subject Classification: 37J06; 37J39; 53D17; 37C86; 70G45; 37C79 1 Introduction The hamiltonization problem on a smooth manifold is the question of whether a dynamical system admits a Hamiltonian formulation. In the framework of Poisson Geometry, given a vector fieldX on a smooth manifoldM , the problem consists in finding a scalar function h and a Poisson structure π on M such that X = π(dh, ·). (1.1) If such a Poisson structure π and function h exist, then X is said to be hamiltonizable on M . The problem of finding a Poisson structure with respect to which a flow of ODEs, or a vector field, is Hamiltonian has received considerable attention from several decades ago. At the earlier stages, this problem concerned to the existence of a Hamiltonian formulation of a set of ODEs and their explicit formulation, mainly in two and three dimensions [1, 6, 16, 17, 32]. Some geo- metric approaches to the hamiltonization problem can be found in the works of Whittaker [32], Perlick [26], S. Hojman [19, 20] and R. Alvarado Flores et al. [2]. A solution to this problem is given by the Lie–Königs theorem [32], which requires the solvability of the ODEs. V. Perlick gives a generalization of the Lie–Konigs theorem in a more global setting, but only when the original system is even-dimensional, so that a time-dependent symplectic structure exists [26]. S. Hojman solves the hamiltonization problem of a vector field by constructing a Poisson struc- ture, provided that it admits an infinitesimal symmetry and a first integral [20]. A geometric reformulation of Hojman results is presented in [2], where the hamiltonization problem is solved for a vector field being in a two-dimensional Lie subalgebra and also admitting a first integral. mailto:misael.avendano@unison.mx mailto:a214200511@unison.mx mailto:jose.ruiz@unison.mx mailto:eduardo.velasco@unison.mx https://doi.org/10.3842/SIGMA.2022.038 2 M. Avendaño-Camacho et al. In recent works, the hamiltonization problem has also been considered, both theoretically [23] and in applied contexts [3]. In this paper, we present a geometric approach to the hamiltonization problem for vector fields (admitting zeroes, in general). In particular, we give positive answers to the hamiltonization problem for vector fields on orientable manifolds with sufficient first integrals, and also for vector fields admitting first integrals that are not necessarily regular. In the latter case, this improves the Hojman construction and the approach of [2]. Furthermore, we provide solutions to the hamiltonization problem for Lie group actions in some particular cases. Our first result deals with vector fields on an m-dimensional manifold admitting m− 1 first integrals. We show that such vector fields are Hamiltonian with respect to m − 1 Poisson structures defined on the open set where the first integrals are independent. More precisely, in Theorem 3.1 we prove that if a vector field X has m − 1 first integrals h1, h2, . . . , hm−1 functionally independent on a open dense set U , then there exist m − 1 Poisson structures π1, . . . , πm−1 on U with rank at most two such that π♯i (dhj) = δijX, i, j = 1, 2, . . . ,m− 1, where δij denotes the kronecker delta. That is, for each i = 1, 2, . . . ,m − 1, X is Hamiltonian with respect to πi with Hamiltonian functions hi and the rest of first integrals are Casimir fuctions of πi. Furthermore, the Poisson structures πi commute with respect to the Schouten– Nijenhuis bracket: [πi, πj ] = 0 for all i, j = 1, . . . ,m − 1. These facts generalize the results presented in [16]. We also describe the dependence of the Poisson structures on the choice of the volume form, which allows us to show in Proposition 3.4 that one can drop the orientability hypothesis. Moreover, we also improve the previous result by relaxing the hypothesis on the number of first integrals: if a vector field X on an m-dimensional manifold admits m − 2 independent first integrals on an open dense set and an invariant volume form, then X is hamiltonizable under a suitable topological condition on the common level sets of the first integrals. This is the content of Theorem 3.5, which in the three-dimensional case recovers the hamiltonization criteria given by Gao in [15] for Lotka–Volterra systems; but the topological hypotheses of our theorem were obviated there. These hypotheses are necessary in general as we exhibit it in Examples 3.9 and 3.10. On the other hand, by considering a correspondence between multivector fields and differential forms on orientable manifolds, we have formulated a criterion of hamiltonization by unimodular Poisson structures in terms of the existence of integrating factors of primitives (Theorem 3.14). In this case, the given vector field is a modular vector field. Finally, Theorem 3.18 is a more general version of Theorem 3.5 that provides a hamiltonization criteria by means of differential forms that are not necessarily product of exact 1-forms: if a vector field preserves a leaf-wise volume of an oriented regular foliation of dimension r, and belongs to the kernel of a nowhere vanishing leaf-wise closed (r−2)-form, then the hamiltonization problem reduces to the triviality of the foliated de Rham cohomology in degree one for the foliation integrating the kernel of the closed form. We also present various results on the hamiltonization problem where orientability is no longer required. Instead, we aim for decomposable Poisson structures. More precisely, in Theorem 4.2, we state necessary and sufficient conditions under which a given vector field X on a manifold M is hamiltonizable by a Poisson structure of the form π = Y ∧X, for some vector field Y on M . In particular, we recover the results of [2, 20], where X must admit a regular first integral. Moreover, following [25], we present in Theorem 4.7 a hamiltonization criteria for vector fields admitting an infinitesimal symmetry and an invariant volume form. On the other hand, in Theorem 4.11 we show that a vector field is hamiltonizable if it admits a first integral and a suitable 2-dimensional foliation. This is a generalization of [12, Theorem 2], where a similar result is proven for the case of nowhere vanishing vector fields with periodic flow. We also present geometric settings in which Theorem 4.2 is applicable: in Theorem 4.13 we show that a vector Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems 3 field is hamiltonizable under suitable conditions involving transversally invariant Riemannian metrics, and in Theorems 4.20 and 4.21 we give hamiltonization criteria for vector fields tangent to the fibers of a submersion. These approaches allowed us to derive Theorem 4.24, where we have shown that the positive answer to the hamiltonization problem for nowhere vanishing vector fields with periodic flow is given by a topological condition: the non-compacity of the manifold. Also, we have derived a hamiltonization criteria for vector fields inducing a proper R-action: such a vector field is hamiltonizable if and only if its orbit space is not compact (Theorem 4.22). Moreover, in Propositions 4.27 and 4.28 we have applied Theorem 4.13 to obtain a hamiltonization criteria for infinitesimal generators of proper actions of general Lie groups. We note that the Poisson structures in Sections 3 and 4 are of rank at most two. This class of Poisson structures have properties that are generally not true for those of higher rank, and that helped us to derive the results of these two sections: they are conformal invariant, are in correspondence to orientable 2-dimensional foliations and are multiple of their nowhere vanishing Hamiltonian vector fields. Moreover, the set of singular points of such structures admits a simple description. Finally, we consider the hamiltonization problem of Lie group actions: given a Lie group action on a manifoldM , whether a Poisson structure onM exists with respect to which the action is Hamiltonian. We want to comment that we have not found in the literature any formulation nor results concerning to the hamiltonization problem of Lie group actions. For the abelian Lie group Tk, we give some conditions under which there exists a bivector field π onM such that the infinitesimal generators are of the form (1.1). If π is Poisson, then the Tk-action is Hamiltonian. In Theorem 5.10 we provide sufficient conditions to solve the hamiltonization problem for 2- dimensional torus actions. The one-dimensional case is the content of Theorem 4.25. 2 Preliminaries A Poisson structure on a smooth manifold M is a bivector field π ∈ Γ ( ∧2TM ) satisfying the Jacobi identity [π, π] = 0. Here, the bracket [−,−] stands for the Schouten–Nijenhuis bracket for multivector fields [10, Section 1.8]. The pair (M,π) is called a Poisson manifold. The rank of a Poisson structure π at a point x ∈ M is defined by rankπx := dimπ♯(T ∗ xM), where the vector bundle map π♯ : T ∗M → TM is given by the usual contraction π♯α := iαπ, for all α ∈ T ∗M . A singular point x of π is characterized by the condition that the rank of π is not constant around x. Otherwise, the point x is said to be a regular point of π. The image π♯(T ∗M) is an integrable distribution on M , called the characteristic distribution of π, and each integral submanifold carries a symplectic structure canonically induced by the Poisson structure π on M . Consequently, the integral symplectic submanifolds define a smooth symplectic foliation of M , which may be singular in general. A vector field X on a Poisson manifold (M,π) is tangent to the symplectic foliation if Xx ∈ π♯(T ∗ xM), for all x ∈ M . Note that for every α ∈ Γ(T ∗M) the vector field X = π♯α is tangent to the symplectic foliation. In particular, if α = dh, for some h ∈ C∞(M), then X is called Hamiltonian vector field. In this case, the function h is said to be a Hamiltonian function for X. We denote by ham(M,π) the Lie algebra of all Hamiltonian vector fields on M . The Casimir functions of π are the Hamiltonian functions c ∈ C∞(M) of the zero vector field, that is, π♯dc = 0. These functions are constant along the leaves of the symplectic foliation ofM . Thus, every tangent vector field is also tangent to the level sets of every Casimir function. In particular, this is true for Hamiltonian vector fields. 4 M. Avendaño-Camacho et al. The infinitesimal Poisson automorphisms of π, or Poisson vector fields, for short, are the vector fields X ∈ Γ(TM) such that LXπ = 0. By poiss(M,π) we denote the Lie algebra of Poisson vector fields, and the Lie subalgebra of tangent Poisson vector fields by poisstan(M,π). We recall that the Lie algebra ham(M,π) is an ideal of both poiss(M,π) and poisstan(M,π). Now, we observe that if π is a Poisson structure and f is an arbitrary function onM , then fπ is not a Poisson structure, in general. We say that π is conformally invariant if fπ is again a Poisson structure on M for any f ∈ C∞(M). A simple computation shows that [fπ, fπ] = −2f π♯df ∧ π. Lemma 2.1. Every Poisson structure with rank at most two is conformally invariant. This lemma follows from the fact that the 3-vector field π♯df ∧ π is a section of the bundle ∧3(π♯(T ∗M)), which is zero due to our rank hypothesis. Tangential Poisson cohomology. Every Poisson structure π onM induces a cochain com- plex, called the Lichnerowicz–Poisson complex. Its cohomology, denoted by H•(M,π), is called the Poisson cohomology of the Poisson manifold (M,π) [24]. It is well-known that H0(M,π) consists of the Casimir functions, the 1-coboundaries are the Hamiltonian vector fields and the 1-cocycles are the Poisson vector fields of (M,π) [10, 24, 28, 29, 30]. So, the cohomology in degree one is the Lie algebra H1(M,π) = poiss(M,π) / ham(M,π). The tangential Poisson cohomology in degree one is the Lie subalgebra H1 tan(M,π) ⊆ H1(M,π) consisting of the Poisson cohomology classes with tangent representatives. More precisely, H1 tan(M,π) := poisstan(M,π) / ham(M,π) (see [14, Definition 2]). Now, recall that the foliated de Rham complex of a regular foliation consists of the graded algebra of foliated differential forms endowed with the foliated exterior derivative. Its cohomology is the foliated de Rham cohomology. Then, we have the following fact [28, Chapter 5]. Proposition 2.2. The tangential Poisson cohomology in degree one of a regular Poisson mani- fold is isomorphic to the foliated de Rham cohomology of its symplectic foliation and, in partic- ular, independent of the leaf-wise symplectic form. In the case when the symplectic foliation is simple, that is, given by the fibers of a submersion, we have the following criterion for the vanishing of the tangential Poisson cohomology in degree one. Proposition 2.3. Let (M,π) be a regular Poisson manifold such that its symplectic foliation is given by the fibers of a submersion. If the fibers are connected and simply connected, then H1 tan(M,π) = 0. This is consequence of Proposition 2.2 and of the fact that the topology of the fibers imply the triviality of the foliated de Rham cohomology in degree one [8, Proposition 7.4]. Orientable Poisson manifolds. Recall that on each oriented m-dimensional manifold M , equipped with a volume form Ω, there exists a one-to-one correspondence between (m − 2)- differential forms and bivector fields given by the following formula: iπΩ = ϱ, π ∈ Γ ( ∧2TM ) , ϱ ∈ Γ ( ∧m−2T ∗M ) . (2.1) Here, the interior product of multivector fields and differential forms is defined by the rule iA∧B = iA ◦ iB, for any A,B ∈ Γ(∧•TM). Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems 5 Consider the (m − 2)-differential form ϱ = 1 2 ∑ i,j ϱijdx 1 ∧ · · · ∧ d̂xi ∧ · · · ∧ d̂xj ∧ · · · ∧ dxm, where the 1-forms with ̂ are omitted. We define the rank of ϱ at a point x ∈M as the rank of the matrix [ϱij(x)]. Here, the set { x1, . . . , xm } is a local coordinate system onM . We also recall that an orientable Poisson manifold is said to be unimodular if admits a volume form invariant under the all Hamiltonian flows [31]. Lemma 2.4. If ϱ is a closed (m − 2)-differential form of rank at most two, then the bivector field π defined by the formula (2.1) is a unimodular Poisson structure on M of rank at most two. Lemma 2.5. Let ϱ be a (m−2)-differential form and suppose that the bivector field π defined by the formula (2.1) is a Poisson structure on M . Then, a vector field X on M is a Hamiltonian vector field, with respect to π, if and only if there exists h ∈ C∞(M) such that iXΩ = dh ∧ ϱ. In particular, a function K ∈ C∞(M) is Casimir for π if and only if dK ∧ ϱ = 0. Finally, one can verify that the Poisson bivector field given by (2.1) is unimodular if and only if ϱ admits an (non-zero) integrating factor a ∈ C∞(M), d(aϱ) = 0. 3 Hamiltonization problem on orientable manifolds We begin by observing that every vector field on an orientable smooth m-dimensional manifold admitting m− 1 independent first integrals is hamiltonizable. Theorem 3.1. Let X be a vector field on an orientable m-dimensional manifold M . Suppose that X admits m − 1 first integrals h1, . . . , hm−1 ∈ C∞(M) which are independent on a open dense subset U ⊆M . Then, there exist unique Poisson structures π1, . . . , πm−1 on U of rank at most two such that π♯idhj = δijX, for all i, j = 1, . . . ,m− 1. (3.1) Moreover, we have [πi, πj ] = 0, for all i and j. Proof. Fix a volume form Ω on M and set β := dh1 ∧ · · · ∧ dhm−1. First, define bivector fields ψi on M by the formula iψi Ω = dh1 ∧ · · · ∧ d̂hi ∧ · · · ∧ dhm−1. (3.2) Since the (m − 2)-form on the right-hand side is closed and of rank at most two on M , from Lemma 2.4 it follows that ψi is a Poisson structure of rank at most two, and hence is conformally invariant due to Lemma 2.1. This implies that we have Poisson structures on U , of rank at most two, defined by πi := (−1)i−1 α(X)ψi. (3.3) Here, α ∈ Γ(T ∗U) is such that Ω = α∧β on U , which exists by the independence of h1, . . . , hm−1. Now, since iXdhj = 0 for all j = 1, . . . ,m − 1, we have i π♯ idhi Ω = dhi ∧ iπiΩ = α(X)β = iXΩ. Hence, π♯idhi = X and π♯idhj = 0 for i ̸= j. Finally, taking into account that iπj iπiΩ = 0 and iπjdiπiΩ = 0, for all i and j, the relations [πi, πj ] = 0 hold. ■ 6 M. Avendaño-Camacho et al. Note that although the ψis in (3.2) are defined on the whole M , we only can assert by construction that the Poisson structures πi in (3.3) exist on the same open set where the first integrals h1, . . . , hm−1 of X are independent. However, in some cases, the Poisson structures πi are well defined on the whole manifold M . This occurs if, for instance, one can choose α in (3.3) such that α(X) is constant, as we illustrate in the following examples. Example 3.2. On R3, oriented with the Euclidean volume form Ω = dx∧dy∧dz, consider the vector field governing the Euler’s rigid body equations X = ayz ∂ ∂x + bxz ∂ ∂y + cxy ∂ ∂z , where a, b and c are nonzero constants related to the principal moments of inertia as follows: a = I2−I3 I2I3 , b = I3−I1 I1I3 and c = I1−I2 I1I2 . The vector fieldX has the following first integrals, consisting of the energy and the square of angular moment: h1 = 1 2 ( x2 I1 + y2 I2 + z2 I3 ) and h2 = 1 2 ( x2 + y2 + z2 ) . Since the first integrals h1 and h2 are independent on U = R3 \ {coordinate axes}, by Theo- rem 3.1, the vector fieldX is Hamiltonian on U with respect to the commuting Poisson structures given by π1 = x ∂ ∂y ∧ ∂ ∂z + y ∂ ∂z ∧ ∂ ∂x + z ∂ ∂x ∧ ∂ ∂y , π2 = x I1 ∂ ∂z ∧ ∂ ∂y + y I2 ∂ ∂x ∧ ∂ ∂z + z I3 ∂ ∂y ∧ ∂ ∂x . However, note that these Poisson structures are well defined on the whole R3, and hence the vector field X is Hamiltonian on R3. Example 3.3. Let T2 be the 2-torus with angular coordinates (φ1, φ2) and volume form Ω = dφ1 ∧ dφ2. For m and n coprime integers, consider the vector field X = ω(nφ1 −mφ2) ( m ∂ ∂φ1 + n ∂ ∂φ2 ) , where ω ∈ C∞(R) is such that ω(t+2π) = ω(t), for all t ∈ R. For every 2π-periodic F ∈ C∞(R), we have that h(φ1, φ2) := F (nφ1 −mφ2) is a first integral of X. Furthermore, if O ⊂ [0, 2π] is the regular domain of F , then U := {(φ1, φ2) \| nφ1 −mφ2 ∈ O} is the regular domain of h. By Theorem 3.1, the vector field X is Hamiltonian with respect to the Poisson structure on U given by π = ω(nφ1 −mφ2) F ′(nφ1 −mφ2) ∂ ∂φ1 ∧ ∂ ∂φ2 . In principle, the Poisson structure π is defined only on the proper open subset U of M . But, if the function ω satisfies the condition ∫ 2π 0 ω(t)dt = 0, then we can choose F as a primitive of ω, F ′ = ω. In this case, the Poisson structure reduces to π = ∂ ∂φ1 ∧ ∂ ∂φ2 , which is well defined on the whole T2 even though the first integral h(φ1, φ2) has singular points. We remark that relations (3.1) imply that each Poisson structure πi has m − 2 Casimir functions given by the first integrals h1, . . . , hi−1, hi+1, . . . , hm−1 of X. Moreover, by (3.3), the rank of πi on U is two except at the points whereX vanishes. Therefore, the set of singular points of the Poisson structure πi on U is the boundary of the open set in which X is non-vanishing. Now, we show that the bivector fields π1, . . . , πm−1 do not depend on the choice of the volume form. For any other volume form Ω on M , there exists α ∈ Γ(T ∗U) such that Ω = Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems 7 α ∧ dh1 ∧ · · · ∧ dhm−1 on U . Since α,dh1, . . . ,dhm−1 are independent on U , we have α = fα + g1dh1 + · · · + gm−1dhm−1, for some f, g1, . . . , gm−1 ∈ C∞(U). Moreover, it holds that Ω = fΩ, and so f is nowhere vanishing. By using α and Ω, let us define ψi and πi analogously as in (3.2) and (3.3), respectively. Consequently, ψi = 1 fψi and, taking into account that dhi(X) = 0, we get πi = (−1)i−1 α(X)ψi = (−1)i−1f α(X) ( 1 f ψi ) = πi. Finally, let us describe the dependence of π1, . . . , πm−1 on the choice of the first integrals h1, . . . , hm−1. If h̃1, . . . , h̃m−1 are (another) first integrals of X, independent on an open dense subset V ⊆M , then dh̃i = ∑m−1 j=1 ajidhj , for some invertible matrix of functions [ aji ] on U ∩ V . Hence, by straightforward computations, the Poisson structures π̃i on V , defined from h̃1, . . . , h̃m−1 by (3.3) and (3.2), satisfy π̃i = m−1∑ j=1 bijπj (3.4) on U ∩ V , where [ bij ] is the inverse of the matrix [ aji ] . The above observations allows us to extend Theorem 3.1 to not necessarily orientable mani- folds. Proposition 3.4. Let M be an m-dimensional manifold. Then, every vector field on M ad- mitting m − 1 first integrals independent on a open dense subset U ⊆ M is Hamiltonian with respect to pairwise commuting Poisson structures on U of rank at most two. Proof. Let X be a vector field onM and h1, . . . , hm−1 ∈ C∞(M) first integrals independent on a open dense subset U ⊆M . Let U = {Uλ}λ∈Λ be a cover of U by orientable open sets. For each λ ∈ Λ, fix a volume form Ωλ on Uλ. By Theorem 3.1, there exist commuting Poisson structures πλ1 , . . . , π λ m−1 on Uλ, of rank at most two, such that ( πλi )♯ dhj = δijX, for i, j = 1, . . . ,m − 1. Now, for fixed λ, λ′ ∈ Λ, we have that πλi and πλ ′ i agree on Uλ ∩ Uλ′ since its constructions are independent of the choice of the volume forms Ωλ and Ωλ′ , respectively. Therefore, there exists πi ∈ Γ ( ∧2TM ) such that πi\|Uλ = πλi , for all λ ∈ Λ. Finally, by construction, it follows that π1, . . . , πm−1 are commuting Poisson structures of rank at most two satisfying (3.1). ■ We remark that if M is non-orientable and connected and X is non-trivial, then the Poisson structures obtained by the conclusion of Proposition 3.4 are singular. Indeed, their singular sets on U are all the boundary of the open set where X is non-vanishing, which is non-empty since M is non-orientable and X admits the maximum number of first integrals. Hamiltonization via invariant volume forms. As an improvement of Theorem 3.1, we proceed to show that we can relax the hypothesis on the number of first integrals: in the case when we have m − 2 independent first integrals of a given vector field X, the existence of an X-invariant volume form is equivalent to the existence of a Poisson structure for which X is an infinitesimal automorphism. Hence, the hamiltonization problem of X turns into providing conditions for the triviality of its Poisson cohomology class. Theorem 3.5. Let X be a vector field on an orientable m-dimensional manifold M admitting m − 2 first integrals c1, . . . , cm−2 ∈ C∞(M) independent on an open dense subset U ⊆ M . Suppose that 1) there exists an X-invariant volume form on M , 2) the level sets of c := (c1, . . . , cm−2) : U ⊆M → Rm−2 are connected and simply connected. 8 M. Avendaño-Camacho et al. Then, the vector field X is Hamiltonian on U with respect to a unimodular Poisson structure on M of rank two on U and zero on M \ U , which admits c1, . . . , cm−2 as Casimir functions. In order to prove this theorem, we need the following: Lemma 3.6. Let c1, . . . , cm−2 ∈ C∞(M) be independent functions on an orientable manifold M of dimensionm. Then, there exists a one-to-one correspondence between volume forms onM and unimodular Poisson structures of rank two on M admitting c1, . . . , cm−2 as Casimir functions. Proof. Let us show that the correspondence between volume forms Ω on M and Poisson struc- tures π of rank two such that each ci is a Casimir function is given by the relation iπΩ = dc1 ∧ · · · ∧ dcm−2. (3.5) Indeed, the correspondence Ω 7→ π follows from Lemmas 2.4 and 2.5. Conversely, suppose we are given a Poisson structure π of rank two with prescribed Casimir functions c1, . . . , cm−2 and fix a volume form Ω0. By Lemma 2.5, we have iπΩ0 ∧ dci = 0, for all i = 1, . . . ,m− 2. Then, it follows from the independence of the cis that iπΩ0 = fdc1 ∧ · · · ∧ dcm−2 for a certain nowhere vanishing f ∈ C∞(M). By setting Ω := 1 fΩ0, the relation (3.5) follows. ■ Remark 3.7. The Poisson structures defined in (3.5) are called Flashcka–Ratiu Poisson struc- tures [7]. Proof of Theorem 3.5. Let Ω be a volume form such that divΩ(X) = 0. By Lemma 3.6, the bivector field π defined by (3.5) is a unimodular Poisson structure on M of rank two on U and zero on M \ U , admitting c1, . . . , cm−2 as Casimir functions. So, the regular foliation of U consisting of the level sets of c : U → Rm−2 is the symplectic foliation of π on U . By the X- invariance of Ω, and since c1, . . . , cm−2 are first integrals of X, we get that X is a tangent Poisson vector field of π on U . Hence, the Poisson cohomology class of X lies in the tangential Poisson cohomology in degree one, [X] ∈ H1 tan(U, π) = poisstan(U, π) / ham(U, π). Since the fibers of c are connected and simply connected, we have H1 tan(U, π) = 0, due to Proposition 2.3. Therefore, the cohomology class of X is trivial and X ∈ ham(U, π). ■ Remark 3.8. The so-called Euler–Jacobi theorem leads to another proof of Theorem 3.5. In- deed, that result implies that X can be solved by quadratures along each fiber of c : U ⊆M → Rm−2 (see, for example, [5] and [21, Theorem 1]). Then, X is locally Hamiltonian along each fiber of c\|U , with respect to the symplectic structure induced by the restriction of π in (3.5). Since the fibers are simply connected, the vector field X is Hamiltonian along every whole fiber of c\|U . Finally, since c : U ⊆ M → Rm−2 is a submersion with connected fibers, there exists a Hamiltonian function h for X with respect to the Poisson structure π [8, Proposition 7.4]. A remarkable difference between the Poisson structures obtained in Theorems 3.1 and 3.5 is that the former encodes the zeroes of the vector field, while latter is always regular, so that the zeroes of the vector field are encoded in the Hamiltonian function. The regularity property in Theorem 3.5, which is due to the independence of the first integrals, is in general a necessary condition, as the following example illustrates. Example 3.9. Consider the functions f, c : R3 → R given by f = 1 2 ( x2 + y2 + z2 ) and c = 2f2. The fibers of c are connected and simply connected since they consist of 2-spheres and a single point. For the Euclidean volume form Ω on R3, the bivector field π defined by iπΩ = dc is a Poisson structure with Casimir function c. Now, consider the vector field X = −y ∂ ∂x + x ∂ ∂y , for which Ω is invariant and c is a first integral. Then, by Theorem 3.5, the vector field X is Hamiltonian for π on the open dense set U := R3 \{0}, where dc is non-vanishing. However, the vector field X is not Hamiltonian for π on the whole R3. In fact, no 1-form α satisfies π♯α = X Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems 9 on R3. To see this, suppose such an α = α1dx + α2dy + α3dz exists. By restricting to U , we get α1\|U = (xz)/ ( 2f2 ) +λxf , α2\|U = (yz)/ ( 2f2 ) +λyf and α3\|U = ( −y2−x2 ) / ( 2f2 ) +λzf , for some λ ∈ C∞(U). The substitution λf = −z/ ( 2f2 ) + µ, for µ ∈ C∞(U), yields α1\|U = xµ, α2\|U = yµ, α3\|U = 1 f + zµ. From the first equation, it follows that α1 vanishes on the yz-plane, so there exists a global ρ ∈ C∞( R3 ) such that α1 = xρ. From here, we have ρ\|U = µ. By the third equation, we get 1 f = (zρ− α3)\|U , which is impossible, because 1/f cannot be smoothly extended to R3. We remark that in the 3-dimensional case, Theorem 3.5 recovers the hamiltonization criteria of [15, first theorem], where the topological hypotheses on the level sets defined by the first integrals were obviated. Let us explain why the topological condition 2 for the fibers in Theorem 3.5 is necessary in general. Suppose we are given a submersion c = (c1, . . . , cm−2) : M → Rm−2 and a fiber-wise 1-form α along the fibers of c satisfying the following properties: (a) α is nowhere vanishing; (b) α is fiber-wise closed; (c) for every nowhere vanishing f ∈ C∞(M), the 1-form fα is not fiber-wise exact. In this setting, for any volume form Ω, the vector field X defined by iXΩ = α∧dc1∧· · ·∧dcm−2, has Ω as invariant volume form, but X cannot be Hamiltonian for a Poisson structure π with Casimir functions c1, . . . , cm−2. In fact, by Lemma 3.6, any two of such Poisson structures differ by multiplication of nowhere vanishing functions. Therefore, if β is a fiber-wise 1-form such that π♯β = X, then β = fα for some nowhere vanishing f ∈ C∞(M). So, by property (c) on α, the vector field X is not Hamiltonian for π. We illustrate this situation in dimension three with a couple of examples. Example 3.10. Consider the vector field X = ∂ ∂z onM = R3\{z-axis}. Clearly, c = 1 2 ( x2+y2 ) is a first integral for X and the Euclidean volume form on R3 is X-invariant. Observe that every nowhere vanishing Poisson structure π with Casimir function c is of the form π = 1 f ( x ∂ ∂y − y ∂ ∂x ) ∧ ∂ ∂z , for a nowhere vanishing function f ∈ C∞(M). Moreover, the vector field X is an infinitesimal Poisson automorphism for π if and only if ∂f ∂z = 0, that is, f = f(x, y). Note that a 1-form α ∈ Γ(T ∗M) satisfies π♯α = X if and only if α = f xdy − ydx x2 + y2 + gdc, for some g ∈ C∞(M). We claim that there is no α of the above form which is exact. Indeed, let γ be the unitary circle on the xy-plane centered at the origin. Then, ∫ γ α = ∫ γ f ̸= 0 since f is nowhere vanishing and γ is connected. In particular, α is not exact. Therefore, X is not a Hamiltonian vector field for Poisson structures with Casimir c. Although the previous example exhibits the necessity of condition 2 in Theorem 3.5, the vector field ∂ ∂z obviously admits a maximal number of independent first integrals and hence is hamiltonizable. In the following example, we present a vector field that admits an invariant volume form, but only a first integral. 10 M. Avendaño-Camacho et al. Example 3.11. Fix λ ∈ R and set F := x2 + y2 + z2 + 1. On R3, the vector field Xλ = (2xz + λy) ∂ ∂x + (2yz − λx) ∂ ∂y + ( 1− x2 − y2 + z2 ) ∂ ∂z (3.6) has the first integral c = ( x2+y2 ) /F 2 and admits the invariant volume form Ω = 1 F 2 dx∧dy∧dz. Moreover, the level surface Ta := c−1(a) is a torus, for a > 1/4. Note that if λ is rational, then the orbits of Xλ along Ta are closed paths and consequently the vector field Xλ admits a second first integral independent with c. By Theorem 3.1, Xλ is Hamiltonizable. If, instead, λ is irrational, then the orbits of Xλ are dense curves on Ta. This implies that any first integral of Xλ is constant on Ta and hence cannot be independent with c. In particular, it cannot be hamiltonized by using Theorem 3.5. Remark 3.12. Even though the vector field Xλ in (3.6) does not admit a second first integral for λ ∈ R \ Q, there exists a Poisson structure with respect to which Xλ is Hamiltonian (see Example 4.30). Now, by using Theorem 3.5, we present families of linear vector fields admitting hamiltoniza- tion. Moreover, we give explicit formulas for the Hamiltonian function, the Poisson structure and the leaf-wise symplectic form based on the following fact: if Π = 1 2Π ij ∂/∂xi ∧ ∂/∂xj is a Poisson structure of rank at most two, then the corresponding leaf-wise symplectic form is given by ωS = − 1 \|Π\|2 Πijdxi ∧ dxj ∣∣ S if dimS = 2, 0 if dimS = 0, (3.7) where \|Π\|2 := ∑ 1≤i<j≤m ( Πij )2 and S is a symplectic leaf of Π. Example 3.13. Let Rn = {x = (x1, . . . , xn)} be the Euclidean vector space. 1. Consider the linear vector field X(x) := Ax · ∂ ∂x on Rn, associated with an n×n matrix A such that trA = 0 and rankA ≤ 2. By the first condition, the canonical volume form on Rn is X-invariant. The second one implies that there exist (linear) independent first integrals c1(x) := v1 · x, . . . , cn−2(x) := vn−2 · x of X, associated with some independent v1, . . . , vn−2 ∈ kerA⊤. Hence, by Theorem 3.5, the vector field X is hamiltonizable on the whole Rn. Furthermore, by (3.7), the Poisson structure in (3.5) is constant and given by π = ∑ 1≤i<j≤n (−1)i+j detP[i,j] ∂ ∂xi ∧ ∂ ∂xj , (3.8) where P[i,j] denotes the (n − 2) × (n − 2) submatrix of P = (v1 · · · vn−2) without the rows i and j. The symplectic foliation consists of the 2-dimensional planes given as the common level sets of the cis equipped with the constant symplectic structure induced by the restriction of ω = ωijdx i ∧ dxj , where ωij = (−1)i+j+1detP[i,j] \|π\|2 , \|π\|2 := ∑ 1≤i<j≤n ( detP[i,j] )2 , 1 ≤ i < j ≤ n. (3.9) In particular, a Hamiltonian function for X is given by h(x) = 1 2x ⊤(WA)x, where W = [ωij ]n×n. Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems 11 2. Let X(x) = Ax · ∂ ∂x be a linear vector field on Rn, associated with a matrix A such that trA = 0 and rankA = 3. Suppose that the non-zero eigenvalues λ1, λ1, λ3 of A⊤ are all real and distinct. Let v1, v2, v3 be eigenvectors of λ1, λ2, λ3, respectively; and v4, . . . , vn a basis of kerA⊤. As in above, the canonical volume form on Rn is X-invariant and the linear functions c4(x) := v4 · x, . . . , cn(x) := vn · x are independent first integrals of X. Moreover, the cubic function c(x) := (v1 ·x)(v2 ·x)(v3 ·x) is also a first integral of X. This follows from LX(vi · x) = λivi · x and trA = 0. Hence, by Theorem 3.5, the vector field X is hamiltonizable on the open set complementary to the union of the 2-planes generated by pairs of v1, v2, v3. By (3.7), the (quadratic) Poisson structure and the corresponding leaf- wise symplectic form are given by (3.8) and (3.9), respectively, with P = (u(x) v4 · · · vn) and u(x) = ∑ cyclic(v1 · x)(v2 · x) v3. A Hamiltonian function for X is given analogously as in above. 3. Fix a, v ∈ R3 such that ∥a∥ > ∥v∥ and a · v = 0. Consider the linear vector field on R4 = { (x, y) \| x ∈ R3, y ∈ R } given by X(x, y) := (a× x+ yv) · ∂ ∂x + (v · x) ∂ ∂y . We show that X can be hamiltonized in two different ways by using Theorem 3.5. First, note that the quadratic function q : R4\{0} → Rs, q(x, y) := 1 2 ( x ·x−y2 ) , is a first integral of X whose level sets q−1(s) are diffeomorphic to S2 × R if s ≥ 0 and to S0 × R3 if s < 0. Now, the hypotheses on a and v imply that there exists w ∈ R3, with ∥w∥ < 1, such that w × a = v. Thus, the linear function ℓ(x, y) := w · x − y is also a first integral of X. Finally, set U := R4 \ {(x, y) \| x− yw = 0}. It follows from the property ∥w∥ < 1 that the level sets of c := (q, ℓ) : U → R2 are diffeomorphic to S2, which is connected and simply connected. Since the Euclidean volume on R4 is X-invariant, Theorem 3.5 implies that X is a Hamiltonian vector field on U with respect to a linear Poisson structure. On the other hand, observe that the linear function φ(x, y) := a · x is also a first integral of X. By restricting to the level sets of (ℓ, φ) : R4 → R2, we get from the first incise that X is Hamiltonian with respect to a constant Poisson structure, with Hamiltonian function h = λ ( x · x− y2 ) , for some constant factor λ (see also [22, Theorem 2]). Hamiltonization via unimodularization. A well-known fact about the hamiltonization problem, which can be seen as a trivial instance of Theorem 3.5, is that a vector field X on a simply connected 2-dimensional manifold is hamiltonizable if and only if it admits an invariant volume form Ω (we recall that every simply connected manifold is orientable). In this case, the Poisson structure π is defined by π♯ = ( Ω♭ )−1 and a Hamiltonian function is a primitive of iXΩ. Motivated by this situation, we formulate the following: Theorem 3.14. Let X be a vector field on an orientable m-dimensional manifold M . Suppose that there exists a volume form Ω such that iXΩ = dϱ, (3.10) for some ϱ ∈ Γ ( ∧m−2T ∗M ) . If ϱ is of rank at most two and admits an integrating factor a ∈ C∞(M), in the sense that d(aϱ) = 0, then X is Hamiltonian on the open set U := {a ̸= 0} with respect to a unimodular Poisson structure of rank at most two on M . Proof. First, by Lemma 2.4, the bivector field π defined by iπΩ = aϱ is a unimodular Poisson structure on M of rank at most two. Now, from the integrating factor property, the function h = 1/a satisfies iXΩ = dh ∧ (aϱ). Hence, by Lemma 2.5, the vector field X is Hamiltonian for π on U , with Hamiltonian function h. ■ 12 M. Avendaño-Camacho et al. Remark 3.15. Observe that Theorem 3.14 is a hamiltonization criteria via “unimodulariza- tion”: the Poisson structure π is unimodular and X is its modular vector field with respect to Ω [31]. Some relations between Hamiltonian vector fields with invariant volume forms and the unimodularity of the corresponding (almost-)Poisson structure in the non-holonomic case are given in [13]. Note that in the particular case when the integrating factor a is nonwhere vanishing, then X is Hamiltonian on the whole M . On the other hand, a necessary condition for (3.10) is that divΩX = 0 holds on M . This is sufficient if the de Rham cohomology of M in degree m− 1 is trivial, for example, if M is an sphere or a real projective space. Corollary 3.16. Let M be a orientable m-dimensional manifold with Hm−1 dR (M) = 0 and X a vector field on M admitting an invariant volume form Ω. Then, condition (3.10) holds for some ϱ. Furthermore, if ϱ admits a nowhere vanishing integrating factor, then X is hamiltoniz- able on M in the following cases: (a) The manifold M is 3-dimensional. (b) The manifold M is 4-dimensional and ϱ ∧ ϱ = 0. This follows from Theorem 3.14 since the items (a) and (b) ensure that ϱ has rank at most two. Example 3.17. Let M be the 3-sphere or the 3-dimensional projective space. Then, for each divergence free vector field X on M , with respect to a volume form Ω on M , there exists a differential 1-form ϱ on M such that (3.10) holds. If ϱ admits a nowhere vanishing integrating factor, thenX is a Hamiltonian vector field onM with respect to a unimodular Poisson structure. Hamiltonization via orientable foliations. Now, we present a generalization of Theo- rem 3.5 in the framework of orientable foliations. Recall that a regular foliation F is said to be orientable if there exists a nowhere vanishing element η ∈ Γ ( ∧topT ∗F ) , called a leaf-wise volume form of F . Theorem 3.18. Let F be an oriented regular foliation on M of dimension r and X ∈ Γ(TM) a vector field tangent to F preserving the leaf-wise volume. Under the following two conditions, the vector field X is hamiltonizable by a unimodular Poisson structure of rank two on M : 1. There exists a nowhere vanishing, leaf-wise closed and locally decomposable leaf-wise (r−2)- form β ∈ Γ ( ∧r−2T ∗F ) such that iXβ = 0. 2. The foliated de Rham cohomology of the foliation Fβ integrating kerβ is trivial in degree one, H1 dR(M,Fβ) = 0. Proof. Let η ∈ Γ(∧rT ∗F) be an X-invariant leaf-wise volume form on F . Define the bivector field π ∈ Γ ( ∧2TF ) by iπη := β. Note that β is of rank at most two since it is locally decompos- able. By the fact that β is leaf-wise closed, we get from Lemma 2.4 that π is Poisson. Moreover, the symplectic foliation of π is precisely Fβ. On the other hand, since iXβ = 0, we get that X is tangent to Fβ and that LXβ = 0. From here and the X-invariance of η, we get that X is an infinitesimal Poisson automorphism for π. Finally, taking into account that H1 dR(M,Fβ) = 0, we conclude from Proposition 2.2 that X is Hamiltonian. ■ Corollary 3.19. Let X be a volume-preserving vector field on an oriented m-dimensional mani- fold M . Then, the vector field X is hamiltonizable by a unimodular Poisson structure of rank two on M if: Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems 13 1. There exist independent 1-forms α1, . . . , αk ∈ Γ(T ∗M) satisfying the integrability condition dαi ∧ α1 ∧ · · · ∧ α̂i ∧ · · · ∧ αk = 0, for all i = 1, . . . , k, and such that X is tangent to the foliation Fα integrating ⋂k i=1 kerαi. 2. There exists a nowhere vanishing, leaf-wise closed and locally decomposable leaf-wise (m− k − 2)-form β ∈ Γ ( ∧m−k−2T ∗Fα ) such that iXβ = 0. 3. The foliated de Rham cohomology of the foliation Fβ integrating kerβ is trivial in degree one, H1 dR(M,Fβ) = 0. Proof. For α := α1 ∧ · · · ∧ αk, the bivector field π defined by iπΩ = α ∧ β is Poisson and X ∈ ham(M,π). ■ As a generalization of Theorem 3.5, the results presented in Theorem 3.18 and Corollary 3.19 allow us to hamiltonize vector fields without having explicit first integrals (see, for instance, Theorem 4.11 and subsequent comments below). Indeed, we can recover Theorem 3.5 by set- ting F as the trivial foliation and β = dc1 ∧ · · · ∧ dcm−2, where c1, . . . , cm−2 ∈ C∞(M) are independent first integrals of the given vector field X. Example 3.20. Consider the manifold M = N × ( R3 \ {y3-axis} ) endowed with the oriented regular foliation (F , η) given by F := ⋃ x∈N {x} × ( R3 \ {y3-axis} ) , η := ea(x) y21 + y22 dy1 ∧ dy2 ∧ dy3 \|R3\{y3-axis}, and the F-tangent and η-preserving vector field on M X = f1(x) y3 ( y1 ∂ ∂y1 + y2 ∂ ∂y2 ) + f2(x, y) ∂ ∂y3 . Here, a, f1 ∈ C∞(N) and f2 ∈ C∞(M) with ∂f2 ∂y3 = 0. Note that the nowhere vanishing and leaf-wise closed differential 1-form β = eb(x) y2dy1 − y1dy2 y21 + y22 , b ∈ C∞(N), is such that iXβ = 0. Moreover, the kernel of β integrates to the subfoliation Fβ of F given by the fibers of the submersion p : M → N × S1, ( x; y1/ √ y21 + y22, y2/ √ y21 + y22 ) . Since the leaves of Fβ are connected and simply connected, we have H1 dR(M,Fβ) = 0 (see [8, Proposition 7.4]). Therefore, by the proof of Theorem 3.18, the vector field X is Hamiltonian on M with respect to the Poisson structure πη,β = −eb(x) ( y1 ∂ ∂y1 + y2 ∂ ∂y2 ) ∧ ∂ ∂y3 . Furthermore, in the case when f2 depends radially on the variables y1, y2, that is, f2(x, y) = g ( x; y21 + y22 ) for some g ∈ C∞(N × R), a Hamiltonian function for X is given by h(x, y) = −1 2e −b(x)[G(x, ln (y21 + y22 )) + y23f1(x) ] , where G ∈ C∞(N × Rt) is such that ∂G/∂t = g ( x, et ) . The case of integrable vector fields in the broad sense. A vector field X on an m- dimensional manifold M is said to be integrable in the broad sense [4, Definition 1] if, for some 0 ≤ k ≤ m − 1, it admits k functionally independent first integrals and a (m − k)-dimensional abelian Lie algebra g ⊆ Γ(TM) of symmetries of X that also preserve the given first integrals. 14 M. Avendaño-Camacho et al. For k = m − 1, a vector field X is integrable in the broad sense if and only if it admits the maximum number of independent first integrals. Indeed, a 1-dimensional Lie algebra of symmetries of X is the one generated by X itself. Now, for k = m − 2, the integrability in the broad sense of X is equivalent to the existence of m− 2 independent first integrals and two independent commutative vector fields preserving X and tangent to the level set of the given first integrals. Observe that the hamiltonization criteria of Proposition 3.4 states that every vector field integrable in the broad sense for k = m − 1 is hamiltonizable. Similarly, the hamiltonization criteria of Theorem 3.5 can be applied for the case k = m− 2. Proposition 3.21. Let X be a vector field on an orientable m-dimensional manifold M that is integrable in the broad sense for k = m− 2. Then: 1. There exists an X-invariant volume form. 2. If the common level sets of the m − 2 independent first integrals of X are connected and simply connected, then X is hamiltonizable on M . Proof. By hypothesis, there exist independent first integrals c1, . . . , cm−2 ofX and commutative vector fields Y1 and Y2 that are symmetries of X and tangent to the level sets of c1, . . . , cm−2. Then, there exists a unique volume form Ω on M such that iY1iY2Ω = dc1 ∧ · · · ∧ dcm−2. Since Y1, Y2 and c1, . . . , cm−2 are all X-invariant, so it is Ω, which proves the item 1. From here and Theorem 3.5, the item 2 follows. ■ 4 Hamiltonization through decomposable Poisson structures In this section, we give necessary and sufficient conditions under which an arbitrary vector fieldX on a smooth manifold M is hamiltonizable via a decomposable Poisson structure π, which is non-regular in general. Specifically, we look for Poisson structures of the form π = Y ∧X, for a vector field Y ∈ Γ(TM). Then, we apply these conditions to give some hamiltonization criteria, involving transversally invariant Riemannian metrics and submersions, to the case of proper actions of 1-dimensional Lie groups and for infinitesimal generators of proper actions of general Lie groups. Consider the open set U ⊆M where the vector field X is non-vanishing. Denote by FX the 1-dimensional foliation of U given by the orbits of X and let νX := TU/TFX be its normal bundle. If X is Hamiltonian with respect to a Poisson structure π of rank at most two, then we have π ∧X ∈ Γ ( ∧3 ( π♯(T ∗M) )) = {0}. This fact, and the nowhere vanishing property of X on U , imply that there exists a vector field Y ∈ Γ(TU) such that π = Y ∧X on U . Moreover, it is clear that π\|U only depends of the class s := [Y ] ∈ Γ(νX) rather than on Y itself. Finally, if h ∈ C∞(M) is a Hamiltonian of X, then dh(Y ) = 1 on U . The properties discussed in the previous paragraph motivate the notion of normal class relative to a vector field X, which encodes the freedom of the choice of the vector field Y such that π = Y ∧X. Definition 4.1. Let X be a vector field onM . A normal class relative to X, or simply a normal class, is an equivalence class of vector fields on M with respect to the relation Y1 ∼X Y2 if and only if (Y2 − Y1) ∧X = 0. Moreover, a normal class [Y ] is said to be � invariant if [X,Y ] ∧X ∧ Y = 0, � normalized with respect to a first integral h of X (or, h-normalized) if dh(Y )X = X. Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems 15 It is straightforward to verify that, for any normal class, the properties of invariance and of h-normalization are independent of the choice of a representative. So, we have the following reformulation for the hamiltonization problem through decomposable Poisson structures. Theorem 4.2. A vector field X is hamiltonizable by a Poisson structure of the form Y ∧ X on M if and only if X admits an invariant normal class that is normalized with respect to a first integral of X that is regular on supp(X). In this case, given a first integral h regular on supp(X), a Poisson structure π on M satisfying π♯dh = X is given by π := { Y0 ∧X on supp(X), 0 elsewhere, (4.1) where Y0 is any vector field defined on a neighborhood of supp(X) satisfying dh\|supp(X)(Y0) = 1 and [X,Y0] ∧X = 0. In other words, the hamiltonization problem for X by decomposable Poisson structures Y ∧X translates into the construction of normalized invariant normal classes. Note that for every Poisson structure of the form π = Y ∧X, for which X is Hamiltonian, it follows from (4.1) that the zeroes of X and π agree. Consequently, rankπp = { 2 if Xp ̸= 0, 0 if Xp = 0, which implies that the set of singular points of π = Y ∧ X is the boundary of the open set in which X is non-vanishing. In particular, the Poisson structure π is regular if and only if, on each connected component of M , the vector field X is nowhere-vanishing or zero. We have divided the proof of Theorem 4.2 into a sequence of lemmas. As a first step, observe that one can associate to each normal class s = [Y ] relative to a vector field X on M a well- defined bivector field πs given by πs := Y ∧X. Lemma 4.3. The assignment s 7→ πs is a one-to-one correspondence between normal classes relative to X and bivector fields decomposable by X. Moreover, (i) the class s is invariant if and only if πs is Poisson, (ii) the class s is h-normalized if and only if π♯sdh = X. Proof. By definition of normal class, the assignment s 7→ πs is injective. For the surjectivity, simply note that the decomposable bivector π = Y ∧X is the image of the normal class defined by Y . On the other hand, if Y is a representative of the normal class s, then [πs, πs] = 2[X,Y ]∧ X ∧ Y , so the Poisson property for πs is equivalent to the invariance of s. Finally, if h is a first integral of X, then we have π♯sdh = dh(Y )X, which implies that s is h-normalized if and only if π♯sdh = X. ■ Now, we show that the first integrals normalizing some normal class of a vector field X are those that are regular on the support of X. Lemma 4.4. Let X be a vector field with first integral h. Then, an h-normalized class relative to X exists if and only if h is regular on supp(X). In this case, if U is the open set where X is non-vanishing, then: 1. Every vector field Y0 defined on a neighborhood of supp(X) satisfying dh\|U (Y0) = 1 deter- mines a unique h-normalized normal class s0 on M . 2. If, in addition, the identity [X,Y0] ∧X ∧ Y0 = 0 holds on U , then s0 is invariant on M . 16 M. Avendaño-Camacho et al. Proof. If Y represents an h-normalized normal class, then dh(Y ) = 1 on U , which implies that dh(Y ) = 1 holds at U = supp(X). Therefore, dh does not vanish on U . Now, suppose that h is regular on U and let Y0 ∈ Γ(TV ) be a vector field defined on a neighborhood V of U such that dh\|U (Y0) = 1. Since U ⊆ V , we can find disjoint open sets W and W ′ satisfying U ⊆ W and M \ V ⊆ W ′, together with a smooth function µ ∈ C∞(M) such that µ\|U = 1 and µ\|M\W = 0. Then, a smooth vector field Y on M is well defined by Y := µY0 on V and Y := 0 on W ′. Since Y agrees with Y0 on U , we have dh(Y )X = X. In other words, the normal class s0 := [Y ] is h-normalized. Finally, since X(µ) = 0 and X vanishes outside U , the invariance of s0 follows from [X,Y0] ∧X ∧ Y0 = 0 on U . ■ Finally, we describe a reformulation of the invariance property for normalized classes. Lemma 4.5. Let X be a vector field on M with first integral h. Let also s be an h-normalized normal class and Y a representative of s. Then, the following assertions are equivalent: 1. The class s is invariant. 2. The identity [X,Y ] ∧X = 0 holds on M . 3. There exists a ∈ C∞(U) such that [X,Y ] = aX holds on U . Here, we denote by U the open subset of M where X is non-vanishing. Proof. Since, by definition, X is nowhere vanishing on U , it readily follows that 2 and 3 are equivalent. Moreover, it is clear that 2 implies 1. So, it is left to show that 1 implies 2. Observe that 0 = idh([X,Y ] ∧ X ∧ Y ) = dh([X,Y ])X ∧ Y − dh(X)[X,Y ] ∧ Y + [X,Y ] ∧ (dh(Y )X) = dh([X,Y ])X ∧ Y + [X,Y ] ∧ X, where in the last step, we have applied that dh(X) = 0 and dh(Y )X = X. Since dh([X,Y ])X ∧ Y = 0 automatically holds on M \ U , it suffices to show that dh([X,Y ]) = 0 on U . From dh(Y )X = X, we get that dh(Y ) = 1 on U . Moreover, by the Koszul’s formula, dh([X,Y ]) = LX(dh(Y ))− LY (dh(X))− d2h(X,Y ). Since dh(X) and dh(Y ) are constant on U , we conclude that dh([X,Y ]) = 0 on U , as desired. ■ Proof of Theorem 4.2. The first assertion follows from Lemmas 4.3 and 4.4. So, it remains to show that π in (4.1) defines a Poisson structure on M satisfying π♯dh = X. To see this, note that π agrees on M with the smooth bivector field πs0 := Y ∧X, where Y ∈ Γ(TM) and s0 are given as in the proof of Lemma 4.4. The fact that πs0 is Poisson and satisfies π♯s0dh = X follows from Lemmas 4.3 and 4.5. ■ Remark 4.6. There is a cohomological approach to the proof of Lemma 4.5, where the non- trivial part is to show that [X,Y ] ∧X ∧ Y = 0 implies [X,Y ] ∧X = 0. Indeed, set π := Y ∧X. Since dh(Y )X = X and [X,Y ] ∧X ∧ Y = 0, we get from Lemma 4.3 that π is Poisson and X is Hamiltonian. In particular, X is an infinitesimal Poisson automorphism for π, which leads to [X,Y ] ∧X = 0. Now, following [25], we use Theorem 4.2 to formulate a hamiltonization criteria of vector fields on orientable manifolds, that are a priori not endowed with a first integral. Theorem 4.7. Let X be a vector field on M admitting an invariant volume form Ω and a vector field Z satisfying [X,Z] = λX, for some λ ∈ C∞(M). Then, the function h := divΩ(Z) − λ is a first integral of X. Moreover, if LZh is non-zero on supp(X), then X is Hamiltonian on M with respect to the Poisson structure of rank at most two defined by π :=  1 LZh Z ∧X on supp(X), 0 elsewhere. (4.2) Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems 17 Proof. First, we show that h is a first integral of X. Indeed, since Ω is X-invariant and Z is a symmetry of X, we have LX(divΩ(Z)Ω) = L[X,Z]Ω + LZ(LXΩ) = dλ ∧ iXΩ + λdiXΩ = (LXλ)Ω+λ(LXΩ) = LX(λΩ), which implies that LX(hΩ) = 0. Moreover, we also have (LXh)Ω = LX(hΩ) − hLXΩ = 0, showing that h is a first integral of X. Now, consider the open set V := {LZh ̸= 0} and the vector field Y0 := 1 LZh Z on V . Then, dh\|V (Y0) = 1 andX∧[X,Y0]∧Y0 = −dh([X,Z]) LZh3 X ∧ (λX) ∧ Z = 0. So, by Lemma 4.4, the vector field Y0 defines a unique normal class for X, which is h-normalized and invariant. Hence, by Theorem 4.2, the vector field X is Hamiltonian on M with respect to the Poisson structure in (4.2). ■ We left for Section 4.1 the discussion of further applications of Theorem 4.2 to the problem of hamiltonization by means of possibly non-regular first integrals. The case of regular first integrals. In this part we review under the light of Theorem 4.2 some known hamiltonization criteria that require the existence of regular first integrals, that is, without critical points. This requirement, however, imposes strong topological conditions on the manifold: it excludes, for instance, a global hamiltonization of vector fields on compact manifolds. As a consequence of Theorem 4.2, we recover the following hamiltonization criteria [2, 20], which implicitly require the regularity of the first integral. Corollary 4.8. Let X be a vector field admitting a regular first integral h. (a) If a vector field Y is such that dh(Y ) = 1 and [X,Y ] = aX, for some a ∈ C∞(M), then π := Y ∧X is Poisson and satisfies π♯dh = X. (b) If a vector field Z is such that dh(Z) is nowhere vanishing and [X,Z] = pX + qZ, for some p, q ∈ C∞(M), then π := 1 dh(Z)Z ∧X is Poisson and satisfies π♯dh = X. Proof. For item (b), it is straightforward to verify that the normal class of Y := 1 dh(Z)Z is invariant and h-normalized. Hence, by Theorem 4.2, the bivector field π := Y ∧X = 1 dh(Z)Z∧X is Poisson and satisfies π♯dh = X. Moreover, item (a) readily follows from (b). ■ The hamiltonization criteria in the item (a) of Corollary 4.8, which is known as the Hojman construction, appears for the first time in [20]. An intrinsic formulation of (a) is found in [2], where statement (b) is presented as a generalization of (a). However, we remark that (b) is not a generalization but a reformulation of (a), since it does not provide new solutions to the hamiltonization problem at all: by Lemma 4.5, every vector field X satisfying (b) also satisfies (a). Indeed, by taking Y := 1 dh(Z)Z, we get dh(Y ) = 1 and [X,Y ] = aX, for a := p/dh(Z). Example 4.9. On M = {x ∈ Rn \| xi ̸= 0, i = 1, . . . , n}, with n ≥ 2, let us consider the vector field X = ∑n i=1 xiFi ∂ ∂xi , where F1, . . . , Fn are homogeneous functions of degree r on M such that F1 + · · · + Fn = 0. Note that the function h(x) = x1 · · ·xn is a regular first integral of X on M . On the other hand, since X is an homogeneous vector field of degree r + 1, we have [X,E] = −rX, where E = x1 ∂ ∂x1 + · · ·+ xn ∂ ∂xn is the Euler vector field. Furthermore, the function dh(E) = nh is nowhere vanishing on M . By Corollary 4.8, the bivector field π = 1 nh ∑ 1≤i<j≤n xixj(Fj − Fi) ∂ ∂xi ∧ ∂ ∂xj , is Poisson and satisfies π♯dh = X. Therefore, X is a Hamiltonian vector field on M . Example 4.10. Let X = Ax · ∂∂x be a linear vector field on Rnx, associated with a real matrix A and E the Euler vector field on Rnx. Observe that [X,E] = 0. 18 M. Avendaño-Camacho et al. � First, suppose that A⊤ admits two distinct real eigenvalues λ1, λ2, with eigenvectors v1, v2, respectively. Fix real numbers r1, r2 satisfying λ1r1 + λ2r2 = 0 and r1 + r2 ̸= 0. Then, on the open set U = {x ∈ Rn \| x · v1 ̸= 0, x · v2 ̸= 0}, the function h(x) = \|v1 · x\|r1 \|v2 · x\|r2 is a regular first integral of X. Moreover, dh(E) = (r1 + r2)h is nowhere vanishing on U . Then, by Corollary 4.8, the vector fieldX is Hamiltonian on U with respect to πh = 1 hE∧X, with Hamiltonian function h. � On the other hand, if the kernel of A⊤ is non-trivial, then for every non-zero v ∈ kerA⊤ the linear function f(x) = v · x is a regular first integral of X. Moreover, df(E) = f is nowhere vanishing on W = {x ∈ Rn \| x · v ̸= 0}. By Corollary 4.8, the vector field X is Hamiltonian on W with respect to πf = 1 fE ∧X, with Hamiltonian function f . Now, we present the following hamiltonization criteria for vector fields admitting a regular first integral and a suitable foliation. This result is a generalization of [12, Theorem 2], where it is formulated for nowhere vanishing vector fields with periodic flow. Theorem 4.11. Let X be a vector field on an m-dimensional manifold M admitting a first integral h ∈ C∞(M) with no critical points. Suppose that X is tangent to a 2-dimensional foliation S transversal to the level sets of h. Then, there exists a unique Poisson structure π = Y ∧X, for some Y ∈ Γ(TM), such that π♯dh = X on M , its symplectic foliation coincides with S on the open set {rankπ = 2} and vanishing at the set {X = 0}. Moreover, on a foliated coordinate chart (U ;x = (x1, . . . , xm−2),y = (y1, y2)) for S, such that y1 = h, we have π\|U = ∂ ∂y1 ∧X. Proof. Since h is transversal to S and has no critical points, around each point in M there exist a foliated coordinate chart (U,x,y) of S such that y1 = h and the leaves of S\|U are the level sets of x. By the fact that X is tangent to S, the functions x1, . . . , xm−2, h are independent first integrals of X. By Theorem 3.1, the bivector field π = ∂ ∂y1 ∧X is the unique Poisson structure on U , of rank at most two, such that π♯dh = X, with Casimir functions x1, . . . , xm−2. Now, for another foliated chart (V, x̃, ỹ) with ỹ1 = h, we have that x̃1, . . . , x̃m−2, h are independent first integrals of X on V . Then, there exists an invertible matrix of functions A = [aji ]i,j=1,...,m−1 on U ∩ V satisfying ajm−1 = δjm−1 and dx̃i = ∑m−2 j=1 ajidxj + am−1 i dh. Since the common level sets of x̃i, x1, . . . , xm−2 are the (2-dimensional) leaves of S\|U∩V , it follows that am−1 i dh ∧ dx1 ∧ · · · ∧ dxm−2 = dx̃i ∧ dx1 ∧ · · · ∧ dxm−2 = 0. This means that the matrix A is of the form ( ∗ 0 0 1 ), with ∗ = [ aji ] i,j=1,...,m−2 . Consequently, its inverse matrix is of the same form. So, by (3.4), the Poisson structure π̃ on V , obtained by Theorem 3.1 using x̃1, . . . , x̃m−2, h, satisfies π̃\|U∩V = π\|U∩V . This shows that π is independent of the choice of the coordinate chart. Hence, the Poisson structure π is global and satisfies π♯dh = X. Furthermore, since x1, . . . , xm−2 are Casimir functions of π, the symplectic foliation of π coincides with S on the open set {rankπ = 2}. Also, by (3.3), the Poisson structure π vanishes at {X = 0}. Finally, by the partition of unity argument, there exists Y ∈ Γ(TM) such that π = Y ∧X. ■ In the case when X has no critical points, one can prove Theorem 4.11 by using the Hojman construction: since the leaves of S are oriented by X and dh\|S , there exists a nowhere vanishing vector field Z ∈ Γ(TS) such that X and Z span TS. Then, the involutivity of S implies that each condition of Corollary 4.8(b) is satisfied. Therefore, X is a Hamiltonian vector field with respect to π = 1/dh(Z)Z ∧X, that clearly satisfies the conclusion of Theorem 4.11. On the other hand, ifM is orientable, then the foliation S is given by the kernel of a differential (m− 2)-form. In this particular case, Theorem 4.11 follows from Theorem 3.18. Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems 19 4.1 Transversally invariant metrics, submersions and proper actions of 1-dimensional Lie groups In this part, we develop some hamiltonization criteria in terms of transversally invariant Rie- mannian metrics (for a more general view of this notion and its geometric applications, see [9, 27]). Also, we consider the hamiltonization problem for vector fields that are vertical on the total space of a submersion, and we derive some results on hamiltonization of infinitesimal generators of proper actions that extend some of the results in [12], where similar approaches are applied in the case of periodic flow vector fields. Hamiltonization via transversally invariant metrics. Let (M, g) be a Riemannian manifold and X a vector field on M . First, we recall that the metric g is said to be X-invariant if LXg = 0. A weaker notion is that of transversal invariance: let O be the orbit of X through a point m ∈ M . Then, the metric η ∈ Γ(T ∗M ⊗ T ∗M) dual to g induces a metric ηO on the annihilator subbundle TO◦. We say that g is transversally invariant if, for eachm ∈M , the map( dmFl t X )∗ : TFltX(m)O ◦ → TmO ◦ induced by the flow FltX of X is an isometry, wherever is well defined. This is equivalent to require that the map dmFl t X : TmM/TmO → TFltX(m)M/TFltX(m)O is an isometry. Infinitesimally, this is just LX\|OηO = 0 for every orbit O of X. It is clear that every X-invariant metric is transversally invariant, but the converse is not true. Lemma 4.12. Let X be a vector field and g a transversally invariant Riemannian metric for X. Then, for every first integral h of X, its gradient ∇h := η♯(dh) commutes with X, and g(∇h,∇h) is a first integral of X. Proof. Since dh is X-invariant, we have [X,∇h] = (LXη) ♯dh. By the transversal invariance, we have β([X,∇h]) = 0 for each 1-form β vanishing on X. Now, let θ be a 1-form orthogonal to those on the annihilator of X. Then, we have that LXθ is orthogonal to the annihilator of X. In particular, θ and LXθ are orthogonal to dh. So, we have θ([X,∇h]) = LX(η(dh, θ))− η(dh,LXθ) = 0. This implies that [X,∇h] = 0. Furthermore, we have LX(g(∇h,∇h)) = LXη(dh,dh) + 2g(LXdh,dh) = 0, where we have used the transversal invariance of g and the invariance of dh. ■ Theorem 4.13. Let X be a vector field on the manifold M that admits 1) a first integral h ∈ C∞(M) that is regular on supp(X), 2) a transversally invariant metric g ∈ Γ(TM ⊗ TM). Then, the vector field X is Hamiltonian on M with respect to the Poisson structure of rank at most two defined by π :=  ∇h g(∇h,∇h) ∧X on supp(X), 0 elsewhere. Proof. Let U and V be the respective open sets where X and dh are nowhere vanishing. Define Y0 ∈ Γ(TV ) by Y0 := ∇h g(∇h,∇h) . Since h is a first integral of X, we have dh ∈ Γ(TO◦) for each orbit O of X. From here, the invariance of h, and the transversal invariance of g, we have from Lemma 4.12 that [X,Y0] = 0. On the other hand, since h is regular on supp(X) and dh(Y0) = 1 on V , we have from Lemma 4.4 that Y0 induces a unique invariant and h-normalized normal class on M . By Theorem 4.2, X is hamiltonizable on M . ■ Now, let us present some remarks on the Hamiltonization problem in the case of invariant metrics. 20 M. Avendaño-Camacho et al. Lemma 4.14. Let X be a vector field and g a transversally invariant Riemannian metric for X on an orientable manifold M . Then, the following assertions are equivalent: (a) The canonical volume form Ωg induced by g is X-invariant. (b) The function g(X,X) ∈ C∞(M) is a first integral of X. (c) The metric g is X-invariant. Proof. Without loss of generality, assume that X is nowhere vanishing. Fix a local orthonor- mal basis of 1-forms θ1, . . . , θm−1, θm such that θ1, . . . , θm−1 is a basis of the annihilator of X. Consider the matrix of functions F = [f ij ] defined by LXθj = ∑m i=1 f i jθi, j = 1, . . . ,m. Note that fmm = 1 2 LX ln(g(X,X)) and fmj = 0 because LX θj vanishes on X for j = 1, . . . ,m−1. Moreover, the transversal invariance property is equivalent to F = ( A 0 0 fmm ) , where A is skew-symmetric. Similarly, the X-invariance of g is equivalent to the skew-symmetry of F . Finally, the canonical volume form is Ωg = θ1 ∧ · · · ∧ θm, so divΩg(X) = ∑m i=1 f i i = fmm . Therefore, (a), (b) and (c) are equivalent to fmm = 0. ■ Note that, in the case of a vector fieldX with an invariant Riemannian metric g, the Laplacian of a first integral is again a first integral. Indeed, let h be a first integral of X. By Lemmas 4.12 and 4.14, the gradient ∇h commutes with X and the canonoical volume form Ωg is X-invariant. Then, by Theorem 4.7, the Laplacian ∆h := divΩg(∇f) is also a first integral of X. In particular, there is a sequence of (not necessarily independent) first integrals h0, h1, . . . recursively defined by h0 := g(X,X) and hn := ∆hn−1. Corollary 4.15. Let X be a vector field, admitting an invariant Riemannian metric g. If the fist integral hn is regular for some n, then X is Hamiltonian on M with respect to π := 1 g(∇hn,∇hn)∇hn ∧X and with Hamiltonian function hn. Hamiltonization of vertical vector fields. In order to present new geometric situations in which one has positive solutions to the hamiltonization problem, we consider the case of (non necessarily regular) vector fields that are tangent to the fibers of a submersion. We begin by considering the case when the given vector field admits an invariant horizontal distribution. Also, we deal with the case when the submersion has 1-dimensional fibers. For instance, this hypothesis has been considered in [12], where it is used to give hamiltonization criteria and examples for nowhere vanishing vector fields with periodic flow. Lemma 4.16. Let p : M → N be a submersion and X ∈ Γ(TM) a vertical vector field. Let f ∈ C∞(N) be regular on an open set N0 containing p(supp(X)), and v ∈ Γ(TN0) such that df \|N0(v) = 1. Then, every vector field Y0 on M0 := p−1(N0) p-related with v satisfying [X,Y0]∧ X = 0 defines a p∗f -normalized and invariant normal class relative to X. Proof. Since Y0 is p-related with v, we have d(p∗f)(Y0) = p∗(df(v)) = 1 on the open set M0 ⊇ supp(X). By the fact that [X,Y0] ∧X = 0 and Lemma 4.4, the result follows. ■ Proposition 4.17. Let M → N be a submersion with vertical distribution V and X ∈ Γ(TV ). If there exists an X-invariant horizontal distribution H, TM = V ⊕ H, and a function f ∈ C∞(N) regular on p(supp(X)), then X is Hamiltonian on M with respect to the Poisson struc- ture π = { Y0 ∧X on supp(X), 0 elsewhere, where Y0 is the horizontal lift of a vector field v defined on the regular domain of f satisfying df(v) = 1. Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems 21 Proof. Let N0 ⊆ N be the open set where f is regular and v ∈ Γ(TN0) such that df(v) = 1. Define Y0 := horH(v) as the horizontal lift of v with respect to H. Since X is vertical, and Y0 is projectable, we have that [X,Y0] is vertical. Denote by prH : TM → H the projection along the splitting TM = V ⊕H. Taking into account that Y0 is horizontal, as well as the X-invariance of H, we have [X,Y0] = [X,prH(Y0)] = prH [X,Y0] = 0. By Lemma 4.16 and Theorem 4.2, we get that X is hamiltonizable on M . ■ Proposition 4.18. Let p : M → N be a submersion of 1-dimensional fibers and X ∈ Γ(TM) a vertical vector field. If there exists f ∈ C∞(N) regular on p(supp(X)), then X is Hamiltonian on M with respect to the Poisson structure π = { Y0 ∧X on supp(X), 0 elsewhere, where Y0 is (any) p-related with a vector field v defined on the regular domain of f satisfying df(v) = 1. Proof. Let N0 ⊆ N be the open set where f is regular and v ∈ Γ(TN0) such that df(v) = 1. Let M0 := p−1(N0) and fix Y0 ∈ Γ(TM0) p-related with v. Since X is vertical, and Y0 is projectable, we have that [X,Y0] is vertical. Taking into account that the p-fibers are 1-dimensional, we get [X,Y0]∧X = 0. By Lemma 4.16 and Theorem 4.2, we have that X is hamiltonizable on M . ■ To end this part, recall that every function on a compact manifold always has a critical point. Furthermore, the converse is also true, in the following sense [18, Theorem 4.8]: Proposition 4.19. Every non-compact manifold admits a smooth function with no critical points. Then, as a direct consequence of this result, as well as of our previous propositions, we get: Theorem 4.20. Let M → N be a submersion over a non-compact manifold N . Then, every vertical vector field admitting an invariant horizontal distribution is hamiltonizable on M . Theorem 4.21. Let M→N be a submersion with 1-dimensional fibers over a non-compact manifold N . Then, every vertical vector field is hamiltonizable on M . If N is non-compact, then there exists f ∈ C∞(N) with no critical points. Hence, Theo- rem 4.20 follows from Proposition 4.17 and Theorem 4.21 from Proposition 4.18. We end this part by observing that the hamiltonization criteria of Theorems 3.1, 4.11 and 4.21 are equivalent. To see this, first recall that Theorem 4.11 is proven by means of Theorem 3.1. On the other hand, Theorem 3.1 can be seen as a particular case of Theorem 4.21. Indeed, given first integrals h1, . . . , hm−1 ∈ C∞(M) of a vector field X on M , independent on an open dense set U ⊆ M , the map p := (h1, . . . , hm−1) : U ⊆ M → Rm−1 is a submersion of 1-dimensional fibers such that X is vertical. Moreover, the coordinate function xi ∈ C∞(Rm−1) is regular and satisfies p∗xi = hi and dxi(vi) = 1 for vi := ∂ ∂xi . By Theorem 4.21, the vector field X is Hamiltonian with respect to a Poisson structure πi of rank at most 2 with Hamiltonian function hi. Also, it is clear that hj is a Casimir function of πi and [πi, πj ] = 0 for j ̸= i. Finally, we observe that Theorem 4.21 is consequence of Theorem 4.11. To see this, fix a regular function f ∈ C∞(N) and a vector field v ∈ Γ(TN) satisfying df(v) = 1. Let Fv be the 1-dimensional foliation of N by the trajectories of v, and S := p∗Fv the 2-dimensional foliation of M whose leaves are the inverse images of the leaves of Fv under p. Note that S is transversal to the level sets of h := p∗f , due to the transversality of Fv to the level sets of f . Moreover, since S contains the p-fibers and X is vertical, it follows that X is tangent to S. Thus, the hypothesis of Theorem 4.11 hold and X is Hamiltonian on M . 22 M. Avendaño-Camacho et al. The rest of this section is devoted to present some hamiltonization criteria that are motivated by Theorem 4.21, but can be also seen as applications of Theorem 4.13. The case of proper actions of 1-dimensional Lie groups. Here we consider the case when the vector field is an infinitesimal generator of a proper action, namely, if it is complete and the action induced by its flow is proper. In this case, we can benefit from the result of Theorem 4.21. Theorem 4.22. Let X be a complete vector field on M such that its flow induces a proper R-action. Then, the following assertions are equivalent: 1. The vector field X is hamiltonizable on M . 2. The orbit space N :=M/R is non-compact. In this case, the vector field X is Hamiltonian on M with respect to the Poisson structure π = Y ∧ X, where Y ∈ Γ(TM) is projectable and satisfies dh(Y ) = 1, for some regular basic function h ∈ C∞(M). Proof. Note that proper actions do not admit compact orbits unless the acting Lie group is compact. Thus, the proper R-action given by the flow of X is also free. So, the vector field X is nowhere vanishing and the orbit space N is a smooth manifold. Therefore, we have a one-to- one correspondence between first integrals h ∈ C∞(M) of X and functions f ∈ C∞(N). Now, suppose that there exist π and h such that π♯dh = X. Then, h is a first integral of X with no critical points and the corresponding f ∈ C∞(N) has no critical points. Thus, N cannot be compact. Conversely, if N is non-compact, then the result follows from Theorem 4.21. ■ Now, we consider the class of vector fields with periodic flow such that their orbits are contained in the fibers of an S1-bundle. This includes vector fields that differ by a first integral scalar factor from the infinitesimal generator of an S1-action. In particular: Lemma 4.23. The trajectories of a nowhere vanishing periodic vector field are the fibers of an S1-bundle. Proof. Let X be a nowhere vanishing vector field on M with periodic flow, ϖ ∈ C∞(M) the period of X and Υ := 1/ϖX. By construction, Υ has period 2π, so it is the infinitesimal generator of a free S1-action on M whose orbits agree with the trajectories of X. ■ With these ingredients we formulate the following: Theorem 4.24. Let X be a nowhere vanishing vector field with periodic flow on M . Then, X is hamiltonizable on M if and only if M is non-compact. In this case, M admits an S1-bundle structure and the vector field X is Hamiltonian on M with respect to the Poisson structure π = Y ∧X, where Y is any projectable vector field on M satisfying dh(Y ) = 1, for some regular basic function h ∈ C∞(M). Proof. IfM is compact, then any function h ∈ C∞(M) has a critical point. Since X is nowhere vanishing, the equation π♯dh = X cannot globally hold on M . Conversely, if M is non-compact, then by Lemma 4.23 we get thatX is a vertical vector field for some S1-bundleM → N . SinceM is non-compact, so is not N . By Theorem 4.21, X is hamiltonizable. ■ The result of Theorem 4.24, is related to the hamiltonization criteria and examples given in [12]. Indeed, Theorem 4.24 characterizes the solvability of the hamiltonization problem for non-vanishing vector fields with periodic flow, improving [12, Theorem 1]. More precisely, we have shown that the fibrating-periodic flow hypothesis is always satisfied, due to Lemma 4.23. Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems 23 Furthermore, by means of Proposition 4.19, the existence of the regular first integral is translated into the non-compacity of the manifold. Finally, we present the following hamiltonization criteria for non-necessarily nowhere vani- shing vector fields with periodic flow. Theorem 4.25. Every vector field X on M with periodic flow admitting a first integral h that is regular on supp(X) is Hamiltonian on M with respect to the Poisson structure π = Y ∧X, where Y = 〈 Ỹ 〉 is the averaging of a vector field Ỹ satisfying dh ( Ỹ ) X = X with respect to the S1-action induced by the flow of X. Proof. By Lemma 4.4, there exists Ỹ ∈ Γ(TM) such that dh ( Ỹ ) X = X. Now, consider the average Y := 〈 Ỹ 〉 of Ỹ under the S1-action induced by the flow of X. Then, we have [X,Y ] = fX, for some f ∈ C∞(M). Moreover, by the S1-invariance of h and X, and the properties of the averaging operator, we get that dh(Y )X = X. In other words, the normal class induced by Y is invariant and h-normalized. By Theorem 4.2, the vector field X is hamiltonizable on M . ■ Observe that the results of this part can be also obtained by means of Theorem 4.13. Indeed, the Poisson structures in Theorems 4.22, 4.24 and 4.25 can alternatively be constructed as π = ∇h g(∇h,∇h) ∧X, where g is any X-invariant Riemmanian metric and h ∈ C∞(M) is a regular first integral of X. The existence of g follows from the properness of the corresponding Lie group action [11, Proposition 2.5.2], and the existence of h in Theorems 4.22 and 4.24 follows from Proposition 4.19 and the non-compactness of the orbit space. Finally, the fact that X is Hamiltonian with respect to π follows from Theorem 4.13. We now illustrate our hamiltonization criteria for periodic vector fields (compare with Ex- ample 3.3). Example 4.26. Consider the torus T2 with natural coordinates (φ1, φ2), φi ∈ R/2πZ. Recall that, for coprime integers m and n, the vector field Υ = m ∂ ∂φ1 + n ∂ ∂φ2 is the infinitesimal generator of an S1-action on T2. Now, fix 2π-periodic functions F, ω ∈ C∞(R), and define X := ω(nφ1 −mφ2)Υ, and h(φ1, φ2) := F (nφ1 −mφ2). Then, the function h is a first integral of X. Moreover, for integers r and s satisfying nr −ms = 1, the vector field Y = r F ′(nφ1 −mφ2) ∂ ∂φ1 + s F ′(nφ1 −mφ2) ∂ ∂φ2 defines an invariant and h-normalized normal class relative to X on its domain. Therefore, X is a Hamiltonian vector field with Hamiltonian function h and with respect to the Poisson structure π := Y ∧X = ω(nφ1 −mφ2) F ′(nφ1 −mφ2) ∂ ∂φ1 ∧ ∂ ∂φ2 . Hamiltonization of infinitesimal generators. Here, we adapt the result of Theorem 4.13 to obtain a hamiltonization criteria for infinitesimal generators of proper actions. 24 M. Avendaño-Camacho et al. Proposition 4.27. Let G be a Lie group with Lie algebra g, acting properly on M . For each ξ ∈ g, let ξM be the infinitesimal generator of the G-action. Suppose that there exists a G- invariant function h ∈ C∞(M) with no critical points. Then, for every G-invariant Riemannian metric g, we have the linear map ξ 7→ πξ := ∇h g(∇h,∇h) ∧ ξM from g to a vector space of Poisson structures of rank at most two on M such that π♯ξdh = ξM . Proof. Since G acts properly on M , there exists an invariant Riemannian metric g on M [11, Proposition 2.5.2]. Now, consider the vector field Y := ∇h g(∇h,∇h) , where ∇h denotes the gradient vector field of h. Clearly, we have dh(Y ) = 1. On the other hand, since h and g are invariant, we get that Y is invariant. In other words, for each ξ ∈ g, we have [ξM , Y ] = 0 and dh(ξM ) = 0. By setting πξ := Y ∧ ξM , the result follows. ■ We also have the following hamiltonization criteria for actions that are proper and free. Proposition 4.28. Let G be a Lie group acting freely and properly on M . If the orbit space M/G is non-compact, then there exists a linear map ξ 7→ πξ := ∇h g(∇h,∇h) ∧ ξM from the Lie algebra g of G to a vector space of Poisson structures of rank at most two on M such that π♯ξdh = ξM . Proof. By Proposition 4.27, it suffices to show that there exists a regular G-invariant function h on M . Since N is non-compact, we have from Proposition 4.19 that there exists f ∈ C∞(M/G) with no critical points. If p : M →M/G is the canonical projection to the orbit space, then the pull-back function h := p∗f is G-invariant, and also regular due to the regularity of f . ■ Remark 4.29. Note that Proposition 4.28 also follows from Theorem 4.20. Indeed, an invariant horizontal distribution always exists because of the G-action is free and proper. We now illustrate the previous ideas on the vector fields of Example 3.11. Example 4.30. Consider the vector fields on R3 X1 = 2xz ∂ ∂x + 2yz ∂ ∂y + ( 1− x2 − y2 + z2 ) ∂ ∂z and X2 = y ∂ ∂x − x ∂ ∂y . Note that X1 and X2 commute and have periodic flow. Therefore, the flows of X1 and X2 induce a T2-action on R3, which is free on M = R3 \ ( {z − axis} ∪ { x2 + y2 = 1, z = 0 }) . Indeed, X1 and X2 have as common first integral h = x2+y2 (x2+y2+z2+1)2 , whose level sets on M are the orbits of the T2-action. Now, let Y0 any vector field on M such that dh(Y0) = 1. Since h is T2-invariant, the averaged vector field Y := ⟨Y0⟩T 2 is invariant and also satisfies dh(Y ) = 1. Note that for each λ ∈ R, the Poisson structure πλ := Y ∧Xλ, where Xλ := X1 + λX2, is such that π♯λdh = Xλ. In other words, the vector fieldXλ is hamiltonizable by Proposition 4.27, regardless of the value of λ. Moreover, recall from Example 3.11 that if λ is irrational, then the orbits of Xλ are dense on the level sets of h, which are compact. Thus, its flow does not define a proper action, and therefore the vector field Xλ cannot be hamiltonized by the result of Theorem 4.22. Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems 25 5 Hamiltonization of torus actions In this section we address the question on hamiltonization of Lie group actions for the particular case in which the acting Lie group is a torus. We adapt some of the ideas developed in Section 4 to provide a Poisson structure so that a given torus action is Hamiltonian with momentum map. Let X1, . . . , Xk ∈ Γ(TM) be infinitesimal generators of an action of Tk on M . Suppose also that we are given Tk-invariant functions h1, . . . , hk ∈ C∞(M) satisfying the following condition: there exist Y1, . . . , Yk ∈ Γ(TM) such that k∑ j=1 dhi(Yj)Xj = Xi, for every i ∈ {1, . . . , k}. (5.1) Remark 5.1. In the case k = 1, condition (5.1) means that h1 is a first integral of X1 that is regular on supp(X1), due to Lemma 4.4. Remark 5.2. By the compactness of Tk, the action admits an invariant Riemaniann metric g. Then, from Lemma 4.14, in the case when X1, . . . , Xk are nowhere vanishing, we have the following family of first integrals hn;i,j := ∆n−1g(Xi, Xj), i, j = 1, . . . , k and n ∈ N, where ∆ is the Laplacian of g, from which we may be able to choose k of them satisfying (5.1). Let also Rk = Lie ( Tk ) be the abelian Lie algebra of Tk. For each ξ = (ξ1, . . . , ξk) ∈ Rk, consider the infinitesimal generator ξM := ∑k i=1 ξiXi and hξ ∈ C∞(M) given by hξ := ∑k i=1 ξihi. Now, define π ∈ Γ ( ∧2TM ) by π := k∑ j=1 Yj ∧Xj . (5.2) Lemma 5.3. For each ξ ∈ Rk, the identity π♯dhξ = ξM holds. In particular, in the case when π is Poisson, the Tk-action on M is Hamiltonian with momentum map J : M → ( Rk )∗ given by J(x)(ξ) := hξ(x). Proof. Since hi is a first integral of Xi, we have from the condition (5.1) that π♯dhi =∑k j=1 dhi(Yj)Xj = Xi. Therefore, π ♯dhξ = ∑k i=1 ξiπ ♯dhi = ∑k i=1 ξiXi = ξM . ■ Now, let us study conditions under which π is a Poisson structure. First, observe that, since the action is abelian, the infinitesimal generators Xi commute, [Xi, Xj ] = 0. Moreover, by averaging with respect to the Tk-action, and taking into account the invariance of each hi and Xi, one can assume that the vector fields Yi given in the condition (5.1) are also invariant, [Xi, Yj ] = 0. Therefore, [π, π] = ∑ 1≤i,j≤k [Yi, Yj ] ∧Xi ∧Xj . Lemma 5.4. Let Uij ⊆M be the open set on which Xi ∧Xj ̸= 0. If, for every i, j ∈ {1, . . . , k}, one has [Yi, Yj ]\|Uij = aijXi + bijXj, for some aij , bij ∈ C∞(Uij), then π is a Poisson structure on M . Actions of 2-dimensional tori. Recall that the Poisson property for π in (5.2) holds in particular for k = 1: this is the content of Theorem 4.25. Now, we focus on the case of an action of a 2-dimensional torus on M , with infinitesimal generators X1, X2 and invariant functions h1, h2 satisfying the condition dhi(Y1)X1 + dhi(Y2)X2 = Xi, i = 1, 2, (5.3) 26 M. Avendaño-Camacho et al. for some T2-invariant Y1, Y2 ∈ Γ(TM). We then have that the bivector field π = Y1 ∧X1 + Y2 ∧X2 (5.4) satisfies [π, π] = [Y1, Y2] ∧ X1 ∧ X2. Now, let U be the open set in which X1 and X2 are independent. Lemma 5.5. For i = 1, 2, we have dhi[Y1, Y2]\|U = 0. In particular, idhi [π, π] = 0. Proof. By (5.3), LYihj = dhj(Yi) = δij . Thus, dhi[Y1, Y2] = LY1LY2hi−LY2LY1hi = 0 on U . ■ Proposition 5.6. Suppose that dimM = 4. Then, the bivector field π in (5.4) is Poisson on M and has rank four on U . Proof. Condition (5.3) implies that h1 and h2 are functionally independent on U , so their common level sets induce a 2-dimensional foliation L of U . Moreover, by the invariance of h1 and h2, the independent vector fields X1 and X2 span TL. On the other hand, by Lemma 5.5, the vector field [Y1, Y2] is tangent to L on U . In particular, the hypothesis in Lemma 5.4 is satisfied, proving that π is Poisson on M . Finally, from the condition (5.3), the vector fields Y1 and Y2 are independent and normal to L on U . Therefore, the vector fields Y1, Y2, X1, X2 are independent and π has maximal rank on U . ■ In general, the orbits of an action are contained in the level sets of invariant functions. In Pro- position 5.6, the dimension hypothesis allows to describe the orbits as level sets of invariant functions. This motives the consideration of the following setting, in which the dimension hypothesis can be relaxed: the orbits and the level sets agree along the leaves of a foliation of the manifold, with an additional transversality condition. Denote by L the foliation of M given by the common level sets of h1 and h2, and by O the foliation of M by the orbits of the T2-action. Definition 5.7. A foliation S of M is said to be compatible with L and O if the following conditions are satisfied: 1. The foliations S and L are transversal, TS + TL = TM . 2. The leaves of the foliation S are invariant, TO ⊂ TS. 3. On the open set U in which X1 ∧X2 ̸= 0, one has TO = TS ∩ TL. Observe that, in the context of Proposition 5.6, such a compatible foliation exists: it consists of the connected components of M . More generally, we have: Proposition 5.8. The bivector field π in (5.4) is Poisson on M if and only if there exists a foliation S on M that is compatible with L and O on U and Y1, Y2 are tangent to S\|U . Proof. By condition (5.3), the invariant functions h1 and h2 are independent on U , as well as Y1 and Y2. In particular, the foliation L is regular on U . Now, suppose that there exists a foliation S on M such that is compatible with L and O on U and Y1, Y2 ∈ Γ(TS\|U ). Then, [Y1, Y2]\|U ∈ Γ(TS). On the other hand, by Lemma 5.5, we have dhi[Y1, Y2]\|U = 0, so [Y1, Y2]\|U ∈ Γ(TL). By the compatibility of S, this implies that [Y1, Y2]\|U ∈ Γ(TO). Finally, since X1 and X2 generate TO, we have [Y1, Y2] ∧ X1 ∧ X2\|U = 0. From here, and the fact that X1 ∧ X2 = 0 outside of U , we conclude that π is Poisson on M . Conversely, suppose that π is Poisson. Let us show that the symplectic foliation S of π is compatible with L and O on U , and satisfies Y1, Y2 ∈ Γ(TS\|U ). By (5.3), we have dhj(Xi) = 0 and dhj(Yi) = δji on U , so the vector fields Y1, Y2, X1, X2 are independent on U . Thus, there exist α1, α2 ∈ Γ(T ∗U) such that αi(Xj) = −δij Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems 27 and αi(Yj) = 0. Then, π♯αi = Yi, so Y1 and Y2 are tangent to S on U . From here, we also have that S and L are transversal on U : TS\|U + TL\|U ⊇ span{Y1\|U , Y2\|U}+ TL\|U = TM \|U Finally, if v ∈ (TS ∩ TL)\|U , then v = a1X1 + a2X2 + b1Y2 + b2Y2, with bi = dhi(v) = 0. So, v ∈ TO, showing that TS ∩ TL = TO. ■ We now give sufficient conditions under which the vector fields Y1 and Y2 in (5.3) can be chosen as sections of a given compatible foliation. Lemma 5.9. Let V be a T2-saturated open set satisfying supp(X1) ∪ supp(X2) ⊆ V and dh1 ∧ dh2 ̸= 0 on V . If S is a compatible regular foliation on V , then there exist T2-invariant vector fields Y1, Y2 ∈ Γ(TM) tangent to S on V and satisfying (5.3). Proof. Since dh1 and dh2 are invariant and independent in the saturated open set V , there exist invariant Ỹ1, Ỹ2 ∈ Γ(TV ) satisfying dhi ( Ỹj ) = δij . Since S and L are regular and transversal on V , there exist Y ′ 1 , Y ′ 2 ∈ Γ(TS) and Z ′ 1, Z ′ 2 ∈ Γ(TL) such that Ỹj = Y ′ j + Z ′ j . Furthermore, since TO ⊆ TS ∩ TL, we have that the action restricts to the leaves of these foliations. So, by averaging and taking into account the invariance of Ỹi, we may assume that Y ′ j , Z ′ j are invariant. Since dhi(Z ′ j) = 0, we have dhi(Y ′ j ) = δij . On the other hand, observe that the closed sets C := supp(X1) ∪ supp(X2) and M \ V are disjoint. Thus, there exist disjoint open sets V ′ andW ′ such that C ⊆W ′ andM \V ⊆ V ′. Let µ ∈ C∞(M) be such that µ\|C = 1 and µ\|V ′ = 0. Since X1, X2 are trivial outside C and µ is constant on C, we have that µ is invariant. Then, there exist smooth, global and invariant vector fields Y1, Y2 ∈ Γ(TM), well-defined by Yj := µY ′ j on V and Yj = 0 on V ′. It is left to show that Y1, Y2 satisfy the condition (5.3). First, observe that the condition is automatically satisfied on M \ C. On the other hand, since Yj agrees with Y ′ j on C ⊆ V , we get that dhi(Yj) = δij on C. Therefore, the condition is satisfied. ■ Theorem 5.10. Let X1 and X2 be infinitesimal generators of a T2-action on a manifold M , and h1, h2 ∈ C∞(M) T2-invariant functions independent on a saturated open neighborhood V of supp(X1)∪ supp(X2). Suppose that there exists a compatible regular foliation S on V , in the sense of Definition 5.7. Then, there exists a Poisson structure π on M such that the T2-action is Hamiltonian on (M,π) with momentum map J : M → (R2)∗ given by J(x)(ξ) := ξ1h1(x) + ξ2h2(x), ξ ∈ R2. Proof. Let U be the open set in which X1 ∧ X2 ̸= 0. By Lemma 5.9, there exist invariant vector fields Y1, Y2 on M tangent to S on V satisfying (5.3). Since S is compatible on U , the bivector field π in (5.4) is Poisson, by Proposition 5.8. Finally, by Lemma 5.3, we get that J is a momentum map. ■ Remark 5.11. We believe that Lemma 5.9, and hence Theorem 5.10, are still true in the more general case when the invariant functions h1, h2 satisfy the condition (5.3), not only when supp(X1) ∪ supp(X2) ⊆ V . However, establishing such fact requires a more profound analysis of the condition (5.3), perhaps by providing an intrinsic formulation of it. 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id	nasplib_isofts_kiev_ua-123456789-211630
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spelling	Avendaño-Camacho, Misael García-Mendoza, Claudio César Ruíz-Pantaleón, José Crispín Velasco-Barreras, Eduardo 2026-01-07T13:41:22Z 2022 Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems. Misael Avendaño-Camacho, Claudio César García-Mendoza, José Crispín Ruíz-Pantaleón and Eduardo Velasco-Barreras. SIGMA 18 (2022), 038, 29 pages 1815-0659 2020 Mathematics Subject Classification: 37J06; 37J39; 53D17; 37C86; 70G45; 37C79 arXiv:2103.00458 https://nasplib.isofts.kiev.ua/handle/123456789/211630 https://doi.org/10.3842/SIGMA.2022.038 Some positive answers to the problem of endowing a dynamical system with a Hamiltonian formulation are presented within the class of Poisson structures in a geometric framework. We address this problem on orientable manifolds and by using decomposable Poisson structures. In the first case, the existence of a Hamiltonian formulation is ensured under the vanishing of some topological obstructions, improving a result of Gao. In the second case, we apply a variant of the Hojman construction to solve the problem for vector fields admitting a transversally invariant metric and, in particular, for infinitesimal generators of proper actions. Finally, we also consider the hamiltonization problem for Lie group actions and give solutions in the particular case in which the acting Lie group is a low-dimensional torus. We are very grateful to the anonymous referees for the observations and suggested improvements on various aspects of this work. This research was partially supported by the Mexican National Council of Science and Technology (CONACYT) under the grant CB2015 no. 258302 and the University of Sonora (UNISON) under the project no. USO315007338. J.C.R.P. thanks CONACyT for a postdoctoral fellowship held during the production of this work. E.V.B. was supported by FAPERJ grants E-26/202.411/2019 and E-26/202.412/2019. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems Article published earlier
spellingShingle	Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems Avendaño-Camacho, Misael García-Mendoza, Claudio César Ruíz-Pantaleón, José Crispín Velasco-Barreras, Eduardo
title	Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems
title_full	Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems
title_fullStr	Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems
title_full_unstemmed	Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems
title_short	Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems
title_sort	geometrical aspects of the hamiltonization problem of dynamical systems
url	https://nasplib.isofts.kiev.ua/handle/123456789/211630
work_keys_str_mv	AT avendanocamachomisael geometricalaspectsofthehamiltonizationproblemofdynamicalsystems AT garciamendozaclaudiocesar geometricalaspectsofthehamiltonizationproblemofdynamicalsystems AT ruizpantaleonjosecrispin geometricalaspectsofthehamiltonizationproblemofdynamicalsystems AT velascobarreraseduardo geometricalaspectsofthehamiltonizationproblemofdynamicalsystems

Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems

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