Generalization of the separation of variables in the Jacobi identities for finite-dimensional Poisson systems
aa r X i v : . [ m a t h - ph ] O c t Generalization of the separation of variables in theJacobi identities for finite-dimensional Poisson systems
Benito Hern´andez-Bermejo Departamento de F´ısica. Escuela Superior de Ciencias Experimentales y Tecnolog´ıa.Universidad Rey Juan Carlos. Calle Tulip´an S/N. 28933–M´ostoles–Madrid. Spain.
Abstract
A new n -dimensional family of Poisson structures is globally characterized and analyzed,including the construction of its main features: the symplectic structure and the reduction tothe Darboux canonical form. Examples are given that include the generalization of previouslyknown solution families such as the separable Poisson structures. PACS codes:
Keywords:
Finite-dimensional Poisson systems — Casimir invariants — Darboux canonicalform — Jacobi identities — Hamiltonian systems. Telephone: (+34) 91 488 73 91. Fax: (+34) 91 664 74 55.E-mail: [email protected] 1 . Introduction
The presence of finite-dimensional Poisson systems in most fields of nonlinear dynamics iswidespread (for instance, see [1] for an overview and a historical discussion). In fact, verydiverse physical systems have been reported to be of the Poisson kind (a sample is given in [2]-[16] and references therein) in spite that such identification often constitutes a nontrivial issue[2, 11, 17, 18]. Thus, the existence of finite-dimensional Poisson systems of applied interestincludes domains such as mechanics, electromagnetism, optics, control theory, fluid mechanics,plasma physics, population dynamics, etc. Recasting a given vector field as a Poisson system(when possible) allows the use of very diverse techniques and specific methods adapted to suchformat, including stability analysis, numerical integration, perturbation methods, bifurcationanalysis and characterization of chaotic behavior, or determination of integability properties andinvariants, just to mention a sample. For instance, see the discussions in [19, 20] for a briefaccount of such application domains and specific methods.In terms of coordinates x , . . . , x n , a finite-dimensional Poisson system is a smooth dynamicalsystem defined in a domain Ω ⊆ R n that can be expressed in the form˙ x ≡ d x d t = J ( x ) · ∇ H ( x ) (1)where x = ( x , . . . , x n ) T and superscript T denotes the transpose of a matrix. Function H ( x )is by construction a time-independent first integral (the Hamiltonian), and the n × n structurematrix J ( x ) is composed by the structure functions J ij ( x ) which are skew-symmetric J ij ( x ) = − J ji ( x ) , i, j = 1 , . . . , n (2)and must be also solutions of the Jacobi PDEs: n X l =1 (cid:0) J li ( x ) ∂ l J jk ( x ) + J lj ( x ) ∂ l J ki ( x ) + J lk ( x ) ∂ l J ij ( x ) (cid:1) = 0 , i, j, k = 1 , . . . , n (3)In addition to the diversity of specific methods and application domains already mentioned,finite-dimensional Poisson systems are of interest because they provide a broad generalizationof classical Hamiltonian systems, allowing for odd-dimensional vector fields and for structurematrices much more general than the classical (constant) symplectic matrices. On the otherhand, Poisson systems maintain a dynamical equivalence to classical Hamiltonian systems, atleast locally, as stated by Darboux’ theorem [1]. The possible rank degeneracy of the structurematrix J ( x ) implies that a certain class of first integrals ( D ( x ) in what follows) termed Casimirinvariants exist. There is no analog in the framework of classical Hamiltonian systems for suchconstants of motion, which are characterized as the solution set of the system of coupled PDEs: J ( x ) · ∇ D ( x ) = 0. The determination of Casimir invariants and their use in order to carry outa reduction (local, in principle) is the cornerstone of the (at least local) dynamical equivalencebetween Poisson systems and classical Hamiltonian systems, as stated by Darboux’ theorem:according to it, the level surfaces of a complete set of Casimir invariants are even-dimensionalmanifolds on which the Poisson system can be reduced to Hamiltonian form, at least in theneighborhood of every point. This justifies that Poisson systems can be regarded, to a largeextent, as a natural generalization of classical Hamiltonian systems. However, the achievement2f the Darboux reduction may be a complicated task in general, specially when it can be carriedout globally. In fact, the global Darboux reduction is known only for a limited number of Poissonstructures and systems [1, 9, 14, 15, 19],[21]-[26].Expressing a given vector field not explicitly written in the form (1) in terms of a Poissonsystem is a nontrivial decomposition to which important efforts have been devoted in the pastyears in a variety of approaches [2]-[14],[16]-[18]. Clearly, the source of the difficulty is twofold:First, a known constant of motion of the system able to play the role of the Hamiltonian isrequired. And second, it is necessary to find a suitable structure matrix for the vector field.Consequently, finding a solution of the Jacobi identities (3) complying also with conditions (2)is unavoidable. This explains the attention deserved in the literature by the obtainment andclassification of skew-symmetric solutions of the Jacobi equations [1, 2, 11, 15],[17]-[26]. Asfar as the Jacobi identities constitute a set of nonlinear coupled PDEs, the characterization ofglobally defined solutions has followed, roughly speaking, a sequence of increasing nonlinearityand increasing dimension (for instance, see [22, 23] for a brief description of different solutionfamilies from the point of view of dimension and nonlinearity). In particular, there is a clearlack of knowledge of solutions verifying the following properties: (i) to be defined for arbitrarydimension n ; (ii) for every n , to allow all possible (even) values of the rank; (iii) to be defined interms of smooth functions of arbitrary degree of nonlinearity; (iv) a complete set of functionallyindependent Casimir invariants can be explicitly determined; (v) it is also possible to constructthe Darboux canonical form; (vi) the previous items can be achieved globally in phase space.The number of known solution families for which all these general conditions hold simultaneouslyis very limited [1, 21, 24].In this work, a new family of skew-symmetric solutions of the Jacobi equations is characterizedand analyzed. Such family generalizes the separable structure matrices [21], a well-known familypresenting the remarkable feature of complying to all the conditions (i)-(vi) just enumerated.Moreover, the new solution family reported in the present work also verifies the six conditionswith full generality. Accordingly, such new family of Poisson structures is not only characterizedbut its global analysis is also carried out in what follows. In addition, it can be seen thatpreviously known types of Poisson structures appearing in a diversity of physical situations nowbecome particular cases of the new solution family, as it will be illustrated in the examples section.The structure of the article is the following. In Section 2 the new family of global Poissonstructures is characterized and its global analysis (including the reduction to the Darboux canoni-cal form) is constructively developed. Section 3 is devoted to the presentation of examples, one ofthem showing the generalization of the family of separable structure matrices. The work finishesin Section 4 with some remarks on the method and possible future outlines.
2. New family of solutions and its global analysis
The fundamental result of this article is presented in this section. As indicated in the Intro-duction, x = ( x , . . . , x n ) T . Similar notation shall be employed for the n -d vectors of variables y = ( y , . . . , y n ) T and z = ( z , . . . , z n ) T . Additionally, in what follows the character w shalldenote a single real variable. Also, for every one-variable real function f ( w ) its derivative will bedenoted f ′ ( w ) and its inverse function f − ( w ), namely f ( f − ( w )) = f − ( f ( w )) = w .3 heorem 1. Let S ≡ ( S ij ) be an n × n skew-symmetric real matrix of rank r ≤ n , let L ≡ ( L ij ) be an n × n invertible real matrix, and let Λ ≡ (Λ ij ) be the inverse of L . For i = 1 , . . . , n , let Ω i ⊆ R be real domains, and define Ω ⊆ R n to be the domain Ω ≡ Ω × . . . × Ω n . In addition,for every i = 1 , . . . , n , let ψ i ( w ) : Ω i → R − { } be a one-variable real function which is C ∞ (Ω i ) and nonvanishing in Ω i . Similarly, for every i = 1 , . . . , n , let λ i ( x ) : Ω → Ω i be the real functiondefined as λ i ( x ) ≡ P nj =1 Λ ij x j . We then have:a) The following set of functions J ij ( x ) = n X k,l =1 L ik S kl L jl ψ k ( λ k ( x )) ψ l ( λ l ( x )) , i, j = 1 , . . . , n (4) constitute the entries of an n × n structure matrix J ( x ) ≡ ( J ij ( x )) globally defined in Ω and of rank r everywhere in Ω .b) Let { k [ r +1] , . . . , k [ n ] } be a basis of Ker { S } , where k [ i ] = ( k [ i ]1 , . . . , k [ i ] n ) T for i = r + 1 , . . . , n .In addition, consider the functions ξ i ( w ) = Z d wψ i ( w ) , i = 1 , . . . , n (5) which are primitives defined in Ω i for which arbitrary integration constants can be chosen.Then for every structure matrix J ( x ) of the form (4) a complete set of Casimir invariantsglobally defined and functionally independent in Ω is given by: D i ( x ) = n X j =1 k [ i ] j ξ j ( λ j ( x )) , i = r + 1 , . . . , n (6) c) Every structure matrix J ( x ) of the form (4) can be constructively and globally reduced in Ω to the Darboux canonical form. Proof.
The proof is constructive. In first place, let us show that there exists a set of n smoothand invertible functions φ i ( w ) for all i = 1 , . . . , n , such that: φ − i ( w ) = ξ i ( w ) = Z d wψ i ( w ) in Ω i , i = 1 , . . . , n (7)In (7) it is assumed that a given (arbitrary) choice of the integration constant is made forevery function φ − i ( w ). Now for all i = 1 , . . . , n , it is ψ i ( w ) = 0 in Ω i , and thus we have (cid:0) φ − i ( w ) (cid:1) ′ = 1 /ψ i ( w ) = 0. Therefore φ i ( w ) exists, and it is unique, smooth and invertible in thedomain Ω ∗ i = φ − i (Ω i ) for every i . Then, using the chain rule for the inverse function we find: ψ i ( w ) = 1 (cid:0) φ − i ( w ) (cid:1) ′ = φ ′ i (cid:0) φ − i ( w ) (cid:1) , i = 1 , . . . , n (8)Taking these elements into account, we now define the following smooth diffeomorphic transfor-mation globally defined in Ω: y i ( x ) = φ − i ( λ i ( x )) , i = 1 , . . . , n (9)4et us now recall the transformation rule of a structure matrix J ( x ) ≡ ( J ij ( x )) after a smoothdiffeomorphism y ≡ y ( x ). The result is a new structure matrix J ∗ ( y ) ≡ ( J ∗ ij ( y )) defined in y (Ω): J ∗ ij ( y ) = n X k,l =1 ∂y i ∂x k J kl ( x ) ∂y j ∂x l , i, j = 1 , . . . , n (10)Now let us apply identities (10) for transformation (9) to matrix (4). We can rewrite (4) as: J ij ( x ( y )) = n X k,l =1 L ik S kl L jl φ ′ k ( y k ) φ ′ l ( y l )In addition, making use again of the chain rule (8) for the inverse function we have: ∂y i ∂x j = Λ ij (cid:0) φ − i (cid:1) ′ ( φ i ( y i )) = Λ ij φ ′ i ( y i )The outcome is then: J ∗ ij ( y ) = n X k,l,r,s =1 Λ ik Λ jl L kr S rs L ls φ ′ i ( y i ) 1 φ ′ j ( y j ) φ ′ r ( y r ) φ ′ s ( y s )After some algebra, this reduces to J ∗ ij ( y ) = S ij , namely J ∗ ( y ) = S . Since S is constant and skew-symmetric, it is in fact a structure matrix defined in R n . As a consequence of diffeomorphism(9), this implies that J ( x ) is also a structure matrix in Ω. In addition, it is by constructionRank {J ( x ) } = Rank { S } = r everywhere in Ω. This proves statement (a) of Theorem 1.Let us now consider statement (b). We take as starting point the structure matrix J ∗ ( y ) = S obtained from J ( x ) after transformation (9). Let { k [ r +1] , . . . , k [ n ] } be a basis of Ker { S } , where k [ i ] = ( k [ i ]1 , . . . , k [ i ] n ) T for i = r + 1 , . . . , n . According to [21], a complete set of Casimir invariantsglobally defined and functionally independent in R n for the structure matrix S is given by: D ∗ i ( y ) = n X j =1 k [ i ] j y j , i = r + 1 , . . . , n (11)Recall that the set of Casimir functions (11) is mapped into a complete set of globally defined andfunctionally independent Casimir invariants D i ( x ) of J ( x ) in Ω according to the transformationrule: D i ( x ) = D ∗ i ( y ( x )) for i = r + 1 , . . . , n . Taking into account (7) and (9), substitution in (11)leads to expression (6) and statement (b) of Theorem 1 is proved.We proceed now to show statement (c). We again take as starting point matrix J ∗ ( y ) = S previously constructed. A second transformation which is also globally defined in R n is performed: z i = n X j =1 P ij y j , i = 1 , . . . , n (12)where P ≡ ( P ij ) is a constant, n × n invertible matrix. According to (10) and (12), the structurematrix J ∗ ( y ) = S is transformed into a new structure matrix J ∗∗ ( z ) ≡ ( J ∗∗ ij ( z )) given by: J ∗∗ ( z ) = P · J ∗ ( y ) · P T = P · S · P T (13)5 classical result of linear algebra [27] shows that matrix P in (13) can always be chosen in sucha way that: J ∗∗ ( z ) = (cid:18) − (cid:19) ( r/ z }| { ⊕ . . . ⊕ (cid:18) − (cid:19) ⊕ O n − r ) z }| { ⊕ . . . ⊕ O (14)where r = Rank { S } is even because S is skew-symmetric, and O ≡ (0) denotes the 1 × J ( x ) has been reduced globally to theDarboux canonical form, since J ∗∗ ( z ) is the direct sum of ( r/
2) symplectic 2 × n − r ) null 1 × O associated with the Casimir invariants, which in the Darbouxrepresentation are decoupled and correspond in (14) to the variables z r +1 , . . . , z n . This showsstatement (c) and completes the proof. ✷ The main presentation of results is thus concluded. In the next section, some applied examplesare analyzed.
3. ExamplesExample 1.
Separable solutions.
Consider the solutions described in Theorem 1 in the particular case in which matrix L is the n × n identity matrix. Taking into account that L ij = Λ ij = δ ij for all i, j = 1 , . . . , n , we see thatnow it is λ i ( x ) = x i for all i and thus, after some direct calculations it is found that solutions(4) are reduced to: J ij ( x ) = S ij ψ i ( x i ) ψ j ( x j ), for i, j = 1 , . . . , n . These are n -dimensional andglobally defined structure matrices of arbitrary rank and degree of nonlinearity. They are termed separable after their functional form resembling the classical separation of variables technique forPDEs. The analytic characterization of separable structure matrices, as well as their symplecticstructure and the global reduction to the Darboux canonical form were presented in [21]. Nowall these features appear as particular cases of Theorem 1. For instance, according to (5-6) wefind that a complete set of independent Casimir invariants is given by D i ( x ) = n X j =1 k [ i ] j Z d x j ψ j ( x j ) , i = r + 1 , . . . , n which is the result found in [21]. Similarly, for the reduction of separable structure matricesto the Darboux canonical form we can apply the general procedure developed in the proof ofTheorem 1. According to such proof, we see from equation (9) that the first transformation tobe performed becomes in the separable case: y i ( x ) = φ − i ( x i ) = Z d x i ψ i ( x i ) , i = 1 , . . . , n (15)Again, diffeomorphism (15) is the one characterized in [21]. Taking the transformation rule (10)into account we arrive to J ∗ ( y ) = S . Then, after this point the reduction to the Darbouxcanonical form carried out in [21] exactly follows the general procedure developed in the proofof Theorem 1. To complete these considerations, it is worth recalling that separable structure6atrices embrace a wide variety of physical systems of interest in the literature, such as Poissonrealizations of Lotka-Volterra and Generalized Lotka-Volterra systems (including examples ofrelevance in both plasma physics and population dynamics), diverse instances of Toda lattices andrelativistic Toda lattices, all constant structure matrices (of which the entire classical Hamiltoniantheory is a particular case), the separable formulation of the Kermack-McKendrick model (seealso Example 2), a differential model associated with circle maps, and Poisson systems arising inthe study of 2 × Example 2.
A nonseparable instance: Kermack-McKendrick model.
In order to illustrate the generality and scope of Theorem 1, it is also interesting to describeone applied instance which is not separable. For this, we shall consider the Poisson formulationof the Kermack-McKendrick model of population dynamics [11, 16]. This system admits a bi-Hamiltonian description in which one of the structure matrices is separable (omitted here for thesake of conciseness, see [21] for details) and thus belongs to the framework considered in Example1. The second structure matrix is not separable, and is the one of interest in what follows: J ( x ) = rx x − − − (16)where r > x i > i = R + − { } for all i = 1 , , , and Ω = Ω × Ω × Ω is the positive 3-d orthant. Thuswe have Rank {J } = 2 in Ω. After some calculations it can be verified that structure matrix (16)is described in terms of Theorem 1 with ψ ( w ) = ψ ( w ) = √ rw , ψ ( w ) = 1, and matrices: S = − , L = − − , Λ = L − = It is worth noting that in this case it is λ ( x ) = x , λ ( x ) = x and λ ( x ) = x + x + x .Accordingly, it is ψ ( λ ( x )) = √ rx , ψ ( λ ( x )) = √ rx and ψ ( λ ( x )) = 1. From this, theconstruction of the structure matrix (16) is straightforward in terms of the elements indicated inTheorem 1. In addition, use of (6) can be made in order to find the only independent Casimirinvariant. For this, we have that a basis of Ker { S } is given by { k [3] } = { (0 , , T } , and ξ ( w ) = w .Consequently, it is D ( x ) = ξ ( λ ( x )) = x + x + x . In addition, for the global reduction to theDarboux canonical form the construction developed in Theorem 1 is again followed. Accordingto (9) the first diffeomorphic transformation to be performed is given by y i ( x ) = φ − i ( λ i ( x )) for i = 1 , ,
3. Following the definition of the φ i ( w ) given in (7), we have φ ( w ) = φ ( w ) = exp( √ rw )and φ ( w ) = w . Consequently, now the diffeomorphic transformation (9) in Ω amounts to: y ( x ) = 1 √ r ln( x ) , y ( x ) = 1 √ r ln( x ) , y ( x ) = x + x + x After some algebra, and taking relationship (10) into account, the transformed structure matrixis J ∗ ( y ) = S , which actually corresponds to the Darboux canonical form (14). We thus see thatthis nonseparable Poisson structure is also described in the framework of Theorem 1.7 . Final remarks In this contribution a new family of Poisson structures has been characterized. Such familycomprises structure matrices of arbitrary dimension and rank, can be globally defined and ana-lyzed, consists of functions of an arbitrary degree of nonlinearity, and is such that a complete setof globally defined independent Casimir invariants can be characterized. Moreover, such solutionfamily can be globally and constructively reduced to the Darboux canonical form. As indicated inthe Introduction, the number of known families of Poisson structures satisfying all such propertiesis very limited. In addition, one of those known families (the separable structure matrices [21])is generalized by the family characterized in Theorem 1. In fact, such generalization is proper(namely the solution family reported in Theorem 1 is not limited to separable solutions, as shownin Example 2). Accordingly, a significant number of physically relevant Poisson structures andsystems becomes unified in the common framework provided by Theorem 1. Therefore thesesystems need not to be considered separately by ad hoc approaches, but inspected in a moregeneral and systematic frame. All these results show that the investigation of Poisson structuresand systems from the point of view of the Jacobi equations provides a fruitful perspective thatdeserves further consideration. 8 eferenceseferences