Generalization of solutions of the Jacobi PDEs associated to time reparametrizations of Poisson systems

Abstract

The determination of solutions of the Jacobi partial differential equations (PDEs) for finite-dimensional Poisson systems is considered. In particular, a novel procedure for the construction of solution families is developed. Such a procedure is based on the use of time reparametrizations preserving the existence of a Poisson structure. As a result, a method which is valid for arbitrary values of the dimension and the rank of the Poisson structure under consideration is obtained. In this article two main families of time reparametrizations of this kind are characterized. In addition, these results lead to a novel application which is also developed, namely the global and constructive determination of the Darboux canonical form for Poisson systems of arbitrary dimension and rank two, thus improving the local result provided by Darboux' theorem for such a case.

Generalized results on the role of new-time transformations in finite-dimensional Poisson systems

“…As indicated, in both cases the result is a Poisson but not a Hamiltonian system (the only exception being the trivial NTT for which η(x) = 1). These are the conclusions that can be obtained by means of the results presented in [3].…”

New global solutions of the Jacobi partial differential equations

“…Thus if we choose in (25) the submatrix composed by the last (n − 2) rows, we see that it is upper triangular with determinant (−1) n−3 (x 2 ) −1 = 0 in Ω. Accordingly, functional independence of D 3 (x), .…”

“…As it can be seen from Theorem 5, together with Theorem 1.e, we have that three-dimensional D ψ -solutions consist essentially of a multiplicative deformation of a constant Poisson structure. In fact, such kind of deformations (based on the multiplication by any Casimir invariant) are closely related to the use of time reparametrizations preserving the Poisson structure [25,27]. This provides a physical interpretation of D ψ -solutions and of the restriction imposed by equations (2,4), at least in the present 3-d context.…”

“…Note also that, in particular, matrix A can be constant. Regarding statement (e) of Theorem 1, it is interesting to note that the multiplicative deformations preserving a Poisson structure have been analyzed in detail in the literature [25,27]. Actually, from these references it is known that every expression of the form η(D r+1 (x), .…”

Generalization of the separation of variables in the Jacobi identities for finite-dimensional Poisson systems

“…Recasting a given vector field as a Poisson system (when possible) allows the use of very diverse techniques and specific methods adapted to such format, including stability analysis, numerical integration, perturbation methods, bifurcation analysis and characterization of chaotic behavior, or determination of integability properties and invariants, just to mention a sample. For instance, see the discussions in [19,20] for a brief account of such application domains and specific methods.…”