A constant of motion in 3D implies a local generalized Hamiltonian structure
aa r X i v : . [ n li n . S I] O c t A constant of motion in 3D implies a localgeneralized Hamiltonian structure
Benito Hern´andez–Bermejo V´ıctor Fair´en Departamento de F´ısica Fundamental, Universidad Nacional de Educaci´ona Distancia. Senda del Rey S/N, 28040 Madrid, Spain.
Abstract
We demonstrate that a Poisson structure can always be associated to ageneral nonautonomous 3D vector field of ODEs by means of a diffeomorphismthat preserves both the orientation and the volume of phase-space. The onlyprerequisite is the existence of one constant of motion.
Keywords:
Ordinary differential equations, generalized Hamiltonian struc-tures, Poisson systems. To whom all correspondence should be addressed. E-mail [email protected] . Introduction
The possibility of associating a generalized Hamiltonian structure [1, 2] (alsoknown in the literature as a Poisson structure) to general flows of ODEs hasdeserved a considerable attention recently, both from the point of view of theexistence of such representations [3] and of their explicit determination in two[4], three [5, 6, 7] and n dimensions [8, 9, 10]. The main advantage of Poissonstructures lies in that they can encompass much wider categories of flows thanthe classical symplectic structure (of which the Poisson structure is just a gen-eralization). However, this is done while preserving the benefits of the classicalsymplectic representation, as proved by Darboux’ theorem [1].The construction of generalized Hamiltonian formulations is of interest inthe field of statistical mechanics, where it has been demonstrated that Nambumechanics [11] is just a particular case of Poisson structure [12, 13, 14]; in theInverse Problem of Mechanics, with results which range from the classical Lie-Koenigs theorem for symplectic structures [15] to the relationship between Pois-son structures and Birkhoff systems [16]; in the discussion of nonlinear stabilityof relative equilibria by the energy-Casimir method [17, 18, 19]; in evolutionarydynamics, in which the connection between Poisson structures and replicatorequations for bimatrix games has been recently established by Hofbauer [20]; inquantization [21], due to the relevance of the link between Poisson and Jacobimanifolds; finally, may we mention the importance in numerical analysis where,in addition to the well-known symplectic integrators [22, 23], a new genera-tion of algorithms is being developed [24, 25], the efficacy of which is ensuredautomatically if the system admits a generalized Hamiltonian description.The Poisson structure for a system of first-order equations˙ x i = v i ( x , . . . , x n , t ) , i = 1 , . . . , n (1)is based: first, on an antisymmetric matrix J whose components J ij are, ingeneral, functions of the x i and satisfy the Jacobi identities J li ∂ l J jk + J lj ∂ l J ki + J lk ∂ l J ij = 0 , (2)where ∂ l means ∂/∂x l , indices i, j, k run from 1 to n and the summation conven-tion over repeated indices is applied; and second, on a Hamiltonian H , which isusually taken to be a time-independent first integral of the system. Then the2ow (1) is said to be Hamiltonian in the generalized sense if it can be writtenas ˙ x i = J ij ∂ j H , i = 1 , . . . , n. (3)In other words, v i can be expressed as v i = [ x i , H ] for the Poisson bracketdefined by [ F, G ] ≡ ( ∂ i F ) J ij ( ∂ j G ) (4)Of course, the task of writting an arbitrary system (1) in the form (3) cannotbe accomplished in general, even if a first integral is known. This is also the casewhen the phase-space is three-dimensional (although this is the single dimensionin which the Jacobi identities (2) are not overdetermined, as pointed out by Haasand Goedert [6]).We shall focus this work on the three-dimensional problem. In this case, ithas been demonstrated that the knowledge of two independent first integrals isalways sufficient to develop a generalized Hamiltonian formulation of the prob-lem [12]. When only one first integral is known, the present-day methods merelyallow the solution of the problem in certain situations [5, 6, 7]. We shall provehere the following general result: the knowledge of a single constant of motionleads locally to a Poisson structure. Given a first integral of the flow (1), theusual approach considers a direct recasting of the system into Poisson form,thus limiting the scope of this format to just those few cases in which the proce-dure leading from (1) to (3) can consistently be completed. Our solution to theproblem does not intend to build the Poisson structure directly on the originalsystem (1) itself, as the previous procedures try to do. We instead overcomeforegoing limitations by showing how an equivalent local Poisson structure can always be associated to an arbitrary three-dimensional system, if we first pro-ceed to submit it to a simple orientation-preserving diffeomorphism, after whichthe hamiltonization procedure is straightforward. This implies automaticallythe strict topologically orbital equivalence between both flows [26]. Moreover,the diffeomorphism is also volume-preserving. The outcome is that it is alwayspossible in practice to exploit the advantages derived from the Hamiltonian for-mulation of the problem, since both systems are dynamically equivalent andinformation can be directly transferred between them due to the invertibitily ofthe transformation. 3 . Poisson structures in 3D flows Due to its ubiquity in all fields of Physics, Chemistry and Biology, 3D sys-tems have received an especial attention in the literature —even an attempt ofgiving a sample of references about the issue would probably exceed the limits ofthis work. This has also been the case with regard to Poisson structures. One ofthe first fundamental results [12] showed that, in general, a 3D system possessesa Poisson structure if two independent first integrals exist. In a subsequent ar-ticle, G¨umral and Nutku [5] reduced the problem to the solution of a nonlinearpartial differential equation, which can be transformed into a Riccati equationif two independent constants of motion are known. Later, Haas and Goedert [6]were able to reformulate the problem as the search of particular solutions of a linear partial differential equation. In their method (which will be our startingpoint) only one time-independent first integral of the system is, in principle,required; however, when two independent constants of motion are known theproblem is reduced to quadrature. This approach was subsequently generalizedby Goedert et al. [7] to include the case of time-dependent, rescalable first inte-grals. To make the list complete, we must also add those approximations whichproceed directly from ansatzs for matrix J , for the Hamiltonian or for both,such as that of Plank [27].The result of these efforts is that it has been possible to construct a Hamil-tonian formulation for some 3D systems (or, at least, for certain integrable orsemi-integrable cases of them). Apart from classical examples such as Euler’sequations for a free rigid body [18, 28] or a particle in a magnetic field [9],we may mention more involved (and interesting) systems like Lotka-Volterraequations [27, 29, 30], the Halphen system [5], Maxwell-Bloch equations [5], theLorenz model [5, 7], the Rabinovich system [7], the RTW interaction [7], andalso some systems of biological relevance such as the Kermac-McKendric modelfor epidemics [31] or the May-Leonard equations [5].Consequently, any generalization of the previously mentioned techniquesleading to the construction of Poisson structures for wider sets of flows wouldlead to an increase of the tools available for their analysis, as follows from theargumentation given in the Introduction. Such a generalization shall be ourgoal in the forthcoming sections of the paper. As mentioned before, our start-ing point will be the technique due to Haas and Goedert, which we briefly recallnow for the sake of completeness. 4et us consider a system of ODEs˙ x i = v i ( x , x , x , t ) , i = 1 , , , (5)for which a time-independent constant of motion H ( x , x , x ) exists. In short,what Haas and Goedert have shown is that, once H is known, a completionof the problem is equivalent to finding one particular solution of the followinglinear partial differential equation, which is essentially a restatement of theJacobi identities (2): v i ∂ i J = AJ + B (6)where A = ∂ i v i − ∂ v i ∂ i H∂ H , B = v ∂ v − v ∂ v ∂ H (7)Once a particular value of J is known, system (5) complies to format (3) withHamiltonian H and structure matrix: J = J , J = v − J∂ H∂ H , J = v + J∂ H∂ H (8)The explicit determination of a solution of (6) for an arbitrary vector field is, inprinciple, not guaranteed. However, as Haas and Goedert point out [6], equation(6) always admits the direct solution J = 0 in the homogeneous case B = 0,though they did not exploit this line of action. We shall do it here. This leadsto a nontrivial solution of the Jacobi equations (2), since J and J do notvanish in general. Further details, such as those related to the preservation ofscale invariance by equations (6) and (7) can be found in the original reference[6].
3. Construction of a Poisson structure for 3D flows
We start by considering a system of the form (5) for which a time-indepen-dent C constant of motion I ( x , x , x ) is known:d I d t = v i ∂ i I = 0 (9)Thus we can write I ( x , x , x ) = I , where I = I ( x (0) , x (0) , x (0)). Weshall assume without loss of generality that ∂ I = 0 (if this is not the case, a5ifferent labeling of the variables can always be adopted in order to exchange x with either x or x , as shown by Haas and Goedert [6]).As emphasized at the end of Section 2, the fundamental equation (6) doesalways admit a solution in the case B = 0 or: v ∂ v − v ∂ v = 0 (10)This identity holds if, in particular: ∂ v = ∂ v = 0 (11)or, in other words, v i ≡ v i ( x , x , t ) for i = 1 ,
2. This suggests the possibility ofperforming a change of dependent variables in system (5) such that (11) holdsfor the target system. This transformation is:˜ x = x ˜ x = x (12)˜ x = I ( x , x , x ) + ϕ ( x , x )where ϕ ( x , x ) is an arbitrary C function. The condition ∂ I = 0 has tworelevant implications. The first is that there exists a unique C function g suchthat the equation I ( x , x , x ) = I can be equivalently written as: x − g ( x , x , I ) = 0 (13)This allows the explicit inversion of the third equation in (12): x = g (˜ x , ˜ x , ˜ x − ϕ (˜ x , ˜ x )) (14)The second implication is that (12) is one-to-one since the Jacobian is not zero.Consequently, transformation (12) is a diffeomorphism.The equations of motion for the transformed system are then:˙˜ x = v (˜ x , ˜ x , g (˜ x , ˜ x , ˜ x − ϕ (˜ x , ˜ x )) , t )˙˜ x = v (˜ x , ˜ x , g (˜ x , ˜ x , ˜ x − ϕ (˜ x , ˜ x )) , t ) (15)˙˜ x = ˙˜ x ˜ ∂ ϕ (˜ x , ˜ x ) + ˙˜ x ˜ ∂ ϕ (˜ x , ˜ x )where ˜ ∂ i denotes ∂/∂ ˜ x i . It is straightforward to check that (15) has the firstintegral: ˜ x − ϕ (˜ x , ˜ x ) = I (16)6his is, in fact, nothing else that the original constant of motion I in terms ofthe new variables, as is evident from the third equation in (12). Then, we canre-express (15) in its final form:˙˜ x = v (˜ x , ˜ x , g (˜ x , ˜ x , I ) , t )˙˜ x = v (˜ x , ˜ x , g (˜ x , ˜ x , I ) , t ) (17)˙˜ x = ˙˜ x ˜ ∂ ϕ (˜ x , ˜ x ) + ˙˜ x ˜ ∂ ϕ (˜ x , ˜ x )This flow verifies condition (11) and it is then a Poisson system. From (16), theHamiltonian is H = ˜ x − ϕ (˜ x , ˜ x ). Notice how the arbitrariness of ϕ remainsas an extra degree of freedom from which we can eventually profit to write theHamiltonian in some desirable form. On the other hand, equations (8) providethe structure matrix: J = v (˜ x , ˜ x , t )0 0 ˜ v (˜ x , ˜ x , t ) − ˜ v (˜ x , ˜ x , t ) − ˜ v (˜ x , ˜ x , t ) 0 (18)where ˜ v i (˜ x , ˜ x , t ) = v i (˜ x , ˜ x , g (˜ x , ˜ x , I ) , t ) for i = 1 ,
2. The reduction to ageneralized Hamiltonian formulation is thus achieved. Transformation (12) isactually global, not dependent on any particular value of the first integral I .On the contrary, the resulting equations of motion (17), and by extension thePoisson structure matrix (18), are particularized to the level surfaces of I , andare thus parametrized by I . Consequently, in order to get (17) and (18), (12)is to be applied locally, in each one of the level surfaces.Taking into account (13), we may choose, in particular, the following repre-sentation for the constant of motion: I ( x , x , x ) = x − g ( x , x , I ) + I (19)Once substituted into (12), the Jacobian of the transformation takes the value1, and it is thus volume-preserving. Since this value is positive, both the originaland the target system are also topologically orbital equivalent [26].
4. Additional considerations
We conclude our exposition by detailing some significant particular situa-tions not considered in the previous section.7
I) Direct substitution of x as a limit case Let us consider transformation (12) in the case in which I is written in theform (19). We can make the following choice for function ϕ : ϕ ( x , x ) = g ( x , x , I ) − I (20)Then the resulting system (17) takes the form:˙˜ x = v (˜ x , ˜ x , g (˜ x , ˜ x , I ) , t )˙˜ x = v (˜ x , ˜ x , g (˜ x , ˜ x , I ) , t ) (21)˙˜ x = ˙˜ x ˜ ∂ g (˜ x , ˜ x , I ) + ˙˜ x ˜ ∂ g (˜ x , ˜ x , I )and the Hamiltonian is H = ˜ x − g (˜ x , ˜ x , I ) + I . Note that this systemis formally the same that would result when using equation (13) to substitutedirectly x in the original equations (5). However, no change of variables isperformed when we proceed in this way. We could say, alternatively, that thefinal variables are just the same than the original ones. The consistency of ourgeneral approach based on a change of dependent variables can then be checkedeasily since equations (12) reduce, as expected, to the identical transformationwhen we substitute (19) and (20) in them. (II) 2D systems without a known first integral It is a well-known result that any 2D flow˙ x i = v i ( x , x , t ) , i = 1 , , (22)possessing a time-independent first integral I ( x , x ) can be written as a Poissonsystem [4, 6]. The analysis of Section 3 allows us to carry out this task whenno first integral is known for equations (22) by means of a one-dimensionalembedding, i.e., by recasting the flow (22) into a three-dimensional one. Thestandard recipe for this purpose consists of the addition of a new variable x tothe system. We do this in the following way:˙ x i = v i ( x , x , t ) , i = 1 , x = v ∂ ϕ ( x , x ) + v ∂ ϕ ( x , x ) (23)where ϕ ( x , x ) is again an arbitrary C function. System (23) is analogous to(17), and it is therefore Hamiltonian in the generalized sense with Hamiltonian8unction H = x − ϕ ( x , x ). On the other hand, (23) is obviously equivalentto the original flow (22) in each level surface H = constant. (III) Quasipolynomial systems Let us consider 3D systems of the quasipolynomial form:˙ x i = x i M i + m X j =1 M ij ( x ) B j ( x ) B j ( x ) B j , i = 1 , , , (24)where m is a positive integer and the rest of coefficients are assumed to be real.These systems have proven to be suitable for representing general nonlinearflows [32, 33]. We shall assume the existence of a first integral I ( x , x , x )verifying the condition ∂ I = 0. If we substitute equations (24) in (10) we find: m X i,j =1 ( M i M j − M j M i ) B j x B i + B j x B i + B j x B i + B j − ++ m X k =1 ( M M k − M M k ) B k x B k x B k x B k − = 0 (25)This leads to the nonlinear system of equations:( M i M j − M j M i )( B j − B i ) = 0 , i, j = 1 , . . . , m ; i = j ( M M k − M M k ) B k = 0 , k = 1 , . . . , m (26)There are two possibilities: i) There is at least one value of i , 1 ≤ i ≤ m , such that B i = 0. Thenequations (26) can be used to demonstrate that there exists a real constant ξ such that (row2) = ξ × (row1) in matrix M . From this it is straightforward toprove that a first integral of the form ( x ) ξ ( x ) − exists. Since this constant ofmotion verifies ∂ [( x ) ξ ( x ) − ] = 0, it must be different to I . Then system (24)has two independent constants of motion and it is integrable. ii) B i = 0 for all i . Then equations (26) are always satisfied.The only nontrivial situation is therefore the second one, which implies, tosum up, that x is not present in the first two equations of system (24). This isprecisely the meaning of equation (11). Consequently, conditions (10) and (11)are completely equivalent, up to trivial and nongeneric cases, for quasipolyno-mial systems of the general form (24). 9 . Final remarks In the preceding sections we have demonstrated how, for a 3D system ofODEs possessing one constant of motion, but for which a generalized Hamilto-nian representation may not exist, such a structure can be achieved when theoriginal vector field is suitably transformed to a completely equivalent system,in such a way that the information obtained (both qualitative and quantitative)from the Hamiltonian formulation can be carried back into the original flow.This approach seems to be relatively new in the literature, since the usual tech-niques try to build the Hamiltonian structure for the original system itself. Theresult is that the problem can be solved only for a limited set of ODEs or, inthe case of the most general approaches, that the practical implementation ofthe Poisson structure is exceedingly difficult. One exception to this trend is thework of Cair´o and Feix [4], where they are able to construct a symplectic struc-ture for 2D flows by means of a rescaling of time. However, it is the authors’conjecture that additional transformations on variables apart from that on timeare necessary in order to handle the problem in three and higher dimensions.In this sense, an appropriate manipulation of the phase-space variables is oneof the most attractive (perhaps the unique) possibility.
Acknowledgements
This work has been supported by the DGICYT (Spain), under grant PB94-0390. B. H. acknowledges a doctoral fellowship from Comunidad Aut´onoma deMadrid. The authors also acknowledge Drs L. Cair´o, F. Haas, M. Plank and G.R. W. Quispel for supplying them with copies of their works.10 eferenceseferences