The Jacobi identities for finite-dimensional Poisson structures: a P.D.E. based analysis of some new constructive results and solution families

Abstract

Jacobi equations constitute a set of nonlinear partial differential equations which arise from the implementation in an arbitrary system of coordinates of a Poisson structure defined on a finite-dimensional smooth manifold. Certain skew-symmetric solutions of such equations are investigated in this dissertation. This is done from a twofold perspective including both the determination of new solution families as well as the construction of new global Darboux analyses of Poisson structures. The most general results investigated refer to the case of solutions of arbitrary dimension. The perspective thus obtained is of interest in view of the relatively modest number of solution families of this kind reported in the literature. In addition, the global Darboux analysis of structure matrices deals, in first place, with the global determination of complete sets of functionally independent distinguished invariants, thus providing a global description of the symplectic structure of phase space of any associated Poisson system; and secondly, with the constructive and global determination of the Darboux canonical form. Such kind of analysis is of interest because the construction of the Darboux coordinates is a task only known for a limited sample of Poisson structures and, in addition, the fact of globally performing such reduction improves the scope of Darboux' theorem, which only guarantees in principle the local existence of the Darboux coordinates. In this work, such reductions sometimes make use of time reparametrizations, thus in agreement with the usual definitions of system equivalence. In fact, time reparametrizations play a significant role in the understanding of the conditions under which the Darboux canonical form can be globally implemented, a question also investigated in detail in this dissertation. The implications of such results in connection with integrability issues are also considered in this context. The dissertation is structured as follows. Chapter 1 is devoted to the revision of diverse classical and well-known results that describe the basic framework of the investigation. The original contributions of the thesis are included in Chapters 2 to 4. Finally, the work ends in Chapter 5 with the presentation of some conclusions.