Separation of variables in the Jacobi identities
aa r X i v : . [ m a t h - ph ] O c t Separation of variables in the Jacobi identities
Benito Hern´andez–Bermejo V´ıctor Fair´en
Departamento de F´ısica Matem´atica y Fluidos, Universidad Nacional de Educaci´on a Distan-cia. Senda del Rey S/N, 28040 Madrid, Spain.
Abstract
A new family of n -dimensional solutions of the Jacobi identities is characterized. Such a familyis very general, thus unifying in a common framework many different well-known Poissonsystems seemingly unrelated. This unification is not only conceptual, but also allows thedevelopment of general global methods of analysis. Keywords:
Finite-dimensional Poisson systems — Jacobi identities — Casimir invariants —Darboux reduction — PDEs. Corresponding author. E-mail: [email protected] . Introduction
Poisson structures [1]–[3] have an important presence in all fields of Mathematical Physics,such as dynamical systems theory [4]–[10], fluid dynamics [11], magnetohydrodynamics [12],optics [13], continuous media [14], etc. Describing a given physical system in terms of aPoisson structure opens the possibility of obtaining a wide range of information which may bein the form of perturbative solutions [15], invariants [9, 16], nonlinear stability analysis [17],bifurcation properties and characterization of chaotic behaviour [13], or integrability results[18], to cite a few.Mathematically, a finite-dimensional dynamical system is said to have a Poisson structureif it can be written in terms of a set of ODEs of the form:˙ x i = n X j =1 J ij ∂ j H , i = 1 , . . . , n, (1)where H ( x ), which is usually taken to be a time-independent first integral, plays the roleof Hamiltonian function. The J ij ( x ) are the entries of an n × n matrix J which may bedegenerate in rank —known as the structure matrix— and they have the property of beingsolutions of the Jacobi identities: n X l =1 ( J li ∂ l J jk + J lj ∂ l J ki + J lk ∂ l J ij ) = 0 (2)In (2), ∂ l means ∂/∂x l and indices i, j, k run from 1 to n . The J ij must also verify theadditional condition of being skew-symmetric: J ij = − J ji for all i, j (3)The possibility of describing a given finite-dimensional dynamical system in terms of a Poissonstructure is still an open problem [4, 6, 7, 8, 10, 19, 20]. The source of the difficulty arises notonly from the need of a known first integral playing the role of the Hamiltonian, but mainlydue to the necessity of associating a suitable structure matrix to the problem. In other words,finding an appropriate solution of the Jacobi identities (2), complying also with the additionalconditions (3), is unavoidable. This explains, together with the intrinsic mathematical interestof the problem, the permanent attention deserved in the literature by the obtainment andclassification of skew-symmetric solutions of the Jacobi identities. In the simplest case ofthree-dimensional flows it has been possible to rewrite equations (2–3) in more manageableforms allowing the determination of some families of solutions [6, 7]. However, this strategyis not applicable when analyzing the general n -dimensional problem (2–3). In such a case,the present-day classification of solutions of (2–3) can be summarized, roughly speaking, as asequence of families of solutions having increasing nonlinearity: constant structures (of whichthe well-known symplectic matrix is just a particular case), linear (i.e. Lie-Poisson) structures[21], affine-linear structures [22] and finally quadratic structures [8, 10, 23, 24].In this letter we present a new family of skew-symmetric, separable solutions of the Jacobiidentities. Due to the unusual generality of such a family —it consists of n -dimensional2olutions not limited to a given degree of nonlinearity— many known Poisson structures appearembraced as particular cases, thus unifying many different systems seemingly unrelated. Aswe shall see, this unification has relevant consequences for the analysis of such systems, sinceit leads to the development of a common framework for the determination of many importantproperties such as the symplectic structure or the Darboux canonical form. Such propertiescan now be characterized globally in a unified and very economic way.The structure of the article is as follows: In Section 2 we establish the main results con-cerning the form of the solutions and their properties. In Section 3 we illustrate the previousresults by means of a number of examples, chosen to show the generality of the solutions andhow very different systems, in principle unrelated, appear now grouped naturally and can beanalyzed in a unified manner. We conclude in Section 4 with some final remarks.
2. Poisson structures arising from separation of variables in the Jacobi identities
Let { ϕ ( x ) , ϕ ( x ) , . . . , ϕ n ( x n ) } be a set of nonvanishing C functions defined on a subsetΩ ⊂ IR n . The need for the nonvanishing condition ϕ i ( x i ) = 0 in Ω will become clear in whatfollows. In the context of this article we define a separable matrix as an n × n matrix definedin Ω which is of the form: S ij = a ij ϕ i ( x i ) ϕ j ( x j ) , a ij ∈ IR , a ij = − a ji for all i, j (4)Obviously, every separable matrix thus defined is skew-symmetric. Moreover, every separablematrix is also a solution of the Jacobi identities (2). Therefore, every separable matrix is astructure matrix. To prove this, we only need to substitute matrix S ij (4) into the Jacobi equations (2). Wefirst arrive to terms of the form: S li ∂ l S jk = a li a jk ϕ l ϕ i ( δ lj ˙ ϕ j ϕ k + δ lk ϕ j ˙ ϕ k ) (5)where ˙ ϕ i means d ϕ i / d x i and δ ij is Kronecker’s delta. Similar expressions are found for theother combinations of indexes. Grouping into (2) and simplifying the deltas we arrive at: n X l =1 ( S li ∂ l S jk + S lj ∂ l S ki + S lk ∂ l S ij ) = ϕ i ϕ j ϕ k [ ˙ ϕ j ( a ji a jk + a ij a jk ) + ˙ ϕ k ( a jk a ki + a kj a ki ) + ˙ ϕ i ( a ki a ij + a ik a ij )] = 0 (6)due to the skew-symmetry of the a ij . This demonstrates one of the main results of this paper.Therefore, in what follows we shall analyze the main properties of the separable structurematrices. According to the previous result, these are all of the form: J ij = a ij ϕ i ( x i ) ϕ j ( x j ) , a ij ∈ IR , a ij = − a ji for all i, j (7)where the ϕ i are nonvanishing arbitrary C functions.We shall start by considering the determination of the Casimir invariants. This can bedone with only the knowledge of the structure matrix, since they are Hamiltonian independent.3n order to solve this problem, the most efficient possibility is to use the abbreviated methoddeveloped by Hern´andez–Bermejo and Fair´en [9]. For this, we first notice that Rank( J ) =Rank( A ) in Ω, where ( A ) ij = a ij . This is a consequence of the nonvanishing character of the ϕ i in Ω. According to [9], this associates naturally the Casimir invariants to the kernel ofmatrix A and leads to the determination of their form: If C = n X j =1 k j Z d x j ϕ j ( x j ) , k = ( k , k , . . . , k n ) T ∈ Ker( A ) (8)where the superscript T denotes the transpose of a matrix, then C is a Casimir function of J .In addition, we have that dim { Ker( A ) } = n − Rank( A ) ≡ m , and thus there are m linearlyindependent vectors that span this kernel. Let { k (1) , k (2) , . . . , k ( m ) } be a basis of Ker( A ), andlet C ( i ) = n X j =1 k j ( i ) Z d x j ϕ j ( x j ) , i = 1 , , . . . , m (9)be the m Casimir invariants associated to this basis. Then, if we evaluate the Jacobianmatrix associated with { C (1) , C (2) , . . . , C ( m ) } it can be immediately seen that such Casimirinvariants are functionally independent. Since m is also the corank of J , this demonstratesthat { C (1) , C (2) , . . . , C ( m ) } is a complete set of independent Casimir functions of the structurematrix (7), any other Casimir invariant being a functional combination of them. Therefore,the symplectic foliation of the separable structure matrices can be completely determined fromthe kernel of the constant matrix A , which is a significant simplification of the problem.We now examine the reduction to the Darboux canonical form. For this, we shall take intoaccount the equation for the transformation of the structure matrix under general diffeomor-phisms, y i = y i ( x ): ˜ J ij ( y ) = n X k,l =1 ∂y i ∂x k J kl ( x ) ∂y j ∂x l (10)We thus introduce the following diffeomorphic transformation, which is globally defined in Ω: y i = Z d x i ϕ i ( x i ) , i = 1 , . . . , n (11)When (7) and (11) are substituted in (10), we obtain:˜ J ij ( y ) = a ij , for all i, j (12)In other words, we have transformed the matrix in such a way that now ˜ J = A is a structurematrix of constant entries. In addition to this, we now perform a second transformation, whichis also globally defined: z i = n X j =1 P ij y j , i = 1 , . . . , n (13)where P is a constant, n × n invertible matrix. According to (10), the structure matrix ˜ J nowbecomes: ˆ J = P · ˜ J · P T = P · A · P T (14)4t is well known [25] that matrix P in (14) can be chosen to have:ˆ J = diag( D , D , . . . , D r/ , ( n − r ) z }| { , . . . ,
0) (15)where r = Rank( A ) is an even number because A is skew-symmetric, and D = D = . . . = D r/ = − ! (16)With (15–16), the structure matrix has been reduced to the Darboux form, since ˆ J is a directsum of 2 × n − r ) null 1 × { z r +1 , . . . , z n } . It is worth emphasizing that the reduction has been completed explicitly andglobally on the domain of interest. This is interesting, since the number of Poisson structuresfor which this can be done is exceedingly limited. Well on the contrary, in the present casethis is possible in a quite natural way.In addition to these advantageous manipulation properties, the separable structure matri-ces embrace and unify many different Poisson structures common in the literature. We shallnow see a sample in the following section.
3. Examples (I) Lotka-Volterra and Generalized Lotka-Volterra systems
The following kind of separable structure matrices J ij = a ij x i x j , a ij ∈ IR , a ij = − a ji for all i, j (17)were first recognized by Plank [8] in the characterization of Poisson structures of the Lotka-Volterra equations, and were important later in the wider case of generalized Lotka-VolterraPoisson systems [10]. However, particular cases of (17) had been previously found in differentcontexts, such as plasma physics [26] or population dynamics [4, 6, 27] (see also the relativisticToda equations in the next example).In (17) we have ϕ i ( x i ) = x i , and therefore the Casimir invariants are immediately foundto be, according to (8), of the form: C = n X j =1 k j ln( x j ) , k = ( k , k , . . . , k n ) T ∈ Ker( A ) (18)In the specific case of Lotka-Volterra equations, the first integrals (18) were already noticedby Volterra himself [28], but they were not generically recognized as Casimir invariants untilPlank’s work [8]. Being Hamiltonian independent, they also appear in more general types ofmodels sharing the structure matrix (17), such as those treated in [10].The first transformation (11) necessary to achieve the Darboux canonical form now is: y i = Z d x i ϕ i ( x i ) = ln( x i ) , i = 1 , . . . , n (19)5he change of variables (19) is to be followed by the linear transformation (13). This kind oftwo-step reduction to a classical Hamiltonian formulation has been known for long —outsidethe framework of Poisson structures— in the particular case of conservative, even-dimensionalLotka-Volterra systems [29]. The realization that such reduction is, in fact, Hamiltonianindependent and inherent to structure matrices of the kind (17) was formalized recently in[10]. (II) Toda lattice and relativistic Toda lattice Toda system, when expressed in Flaschka’s variables [30] ( α , . . . , α N − , β , . . . , β N ) is aPoisson system with brackets { α i , β i } = − α i , { α i , β i +1 } = α i , while the rest of the elementarybrackets vanish. This Poisson bracket corresponds to a separable structure given by: ϕ i ( α i ) = α i , i = 1 , . . . , N − ϕ j ( β j ) = 1 , j = 1 , . . . , N (20) A = O ( N − × ( N − M ( N − × N − M TN × ( N − O N × N ! (21)where the subindexes of the submatrices indicate their sizes, O denotes the null matrix and M ( N − × N = − . . . − . . . . . . − (22)It is immediate that the kernel of A is one-dimensional, a basis of which is provided by thevector k = ( N − z }| { , . . . , , N z }| { , . . . , T . Consequently, from (8) there is only one independentCasimir invariant, C = P Nj =1 R d β j = P Nj =1 β j , which is the result found in [30]. The reductionto the Darboux form also becomes straightforward, since we have to perform transformation(11) ˜ α i = ln( α i ) , i = 1 , . . . , N − β j = β j , j = 1 , . . . , N (23)and then carry out the linear change of variables (13).Analogously, we consider now the relativistic Toda equations expressed in similar variables[30] ( α , . . . , α N − , β , . . . , β N ). Again, it is a Poisson system with brackets { α i , α i +1 } = α i α i +1 , { α i , β i } = − α i β i , { α i , β i +1 } = α i β i +1 , while the rest of the elementary bracketsvanish. This Poisson bracket corresponds to a separable structure of the form (17) examinedin Example I. Therefore, all the considerations made there hold in this context. (III) Further examples: Constant structure matrices, Kermack-McKendric model, circle maps, × games We end the present section with a brief enumeration of other examples which have alsodeserved some interest in the literature. We shall not elaborate on them with the detail of theprevious instances, but only outline the most interesting features.6he first example will be that of the constant skew-symmetric structure matrices, whichappear frequently in very diverse developments. In this case we would like to recall that ϕ i ( x i ) = 1, and thus the Casimir functions (8) are linear. Notice also that transformation(11) reduces to the identity, and therefore the reduction to classical Hamiltonian form onlyinvolves a linear transformation (13–15). These are well-known facts that now appear as asimple particular case.As a second example we touch upon the Kermack-McKendric model [5, 6], which admitsa Poisson structure in terms of matrix: J = − rx x rx x − ax ax (24)where the x i denote the system variables and { a, r } are real constants. We again have aseparable matrix with { ϕ ( x ) , ϕ ( x ) , ϕ ( x ) } = { x , x , } . Therefore, this example turnsout to be very similar to the nonrelativistic Toda lattice examined before, as it can be seenfrom (20–21). We thus find that seemingly unrelated problems can be analyzed in a similar,unifying framework.Next we shall mention the Poisson structures appearing in the study of certain circle maps[6], in which we have: J = − ( x ) ( x ) − ( x ) ( x ) ( x ) ( x ) ( x ) ( x ) (25)We thus have ϕ i ( x i ) = ( x i ) , a more nonlinear kind of function. The evaluation of the Casimirinvariants and the Darboux canonical form do not present any special difficulty in this caseand are omitted.Finally we shall consider a very different kind of structure matrix found in the context of2 × J = x (1 − x ) x (1 − x ) − x (1 − x ) x (1 − x ) 0 ! (26)Now we have that Ω = int { S × S } is the interior of the product of two probability simplices,and ϕ i ( x i ) = x i (1 − x i ). Obviously there are no Casimir functions in this case. Because now A is the 2 × y i = Z d x i x i (1 − x i ) = ln x i − x i ! , i = 1 , ϕ i ( x i ) are nonvan-ishing in the domain Ω. 7 . Final remarks In this letter we have presented a new family of skew-symmetric solutions of the Jacobiidentities which exhibits a number of interesting properties: • It can be found by applying the classical method of separation of variables. It is math-ematically remarkable that such a technique, in principle restricted to the domain ofcertain linear PDEs, can be applied in this context. • The resulting solutions are valid for the n -dimensional problem (2–3), independently ofthe value of n , and are not limited to a given degree of nonlinearity. Well on the contrary,they can contain polynomials of arbitrary degree and general functions as well. • They generalize already known solutions. In fact, they unify in a common frameworkmany different Poisson structures and well-known systems which seemed to be unrelated,and now appear as particular cases of a common family. We have provided a list of themin the examples.And last but not least: • This unification is not only conceptual, but allows the development of general methodsof analysis simultaneously valid for every particular case. Specifically, it is possible todetermine in an algorithmic and explicit way the Casimir invariants and the Darbouxcanonical form. Moreover, these results hold globally in the phase-space —in contrastwith the usual scope of Darboux’ theorem [3], which only guarantees a local reduction.As we have seen, the direct search of new Poisson structures as solutions of the Jacobiidentities leads to a unifying perspective of very diverse, and seemingly unrelated, systems.This is useful not only from a classification point of view —from which we have obtained aremarkably general family of solutions— but also because it is possible, as we have demon-strated, to develop a unified approach for their analysis which becomes much more economicand elegant than a heuristic, case-by-case strategy. Obviously, it is feasible to establish suchtechniques with regard to all Hamiltonian independent properties, such as those treated above.For these reasons, it is the authors’ impression that this kind of analysis of the Jacobiequations can produce interesting results for further generalization and unification of finite-dimensional Poisson structures. In particular, two clear lines of investigation are an additionalgeneralization of the separable solutions (7) found in this paper, as well as the search ofcompletely different families of structure matrices. These possibilities will be the subject offuture research.
Acknowledgements
The authors wish to acknowledge support from the European Union (Esprit WG 24490).8 eferenceseferences