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dc.contributorGarcía, I. A. (Isaac A.)
dc.contributorUniversitat de Lleida. Departament de Matemàtica
dc.creatorHernández Bermejo, Benito
dc.descriptionJacobi 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.
dc.publisherUniversitat de Lleida
dc.rightsADVERTIMENT. L'accés als continguts d'aquesta tesi doctoral i la seva utilització ha de respectar els drets de la persona autora. Pot ser utilitzada per a consulta o estudi personal, així com en activitats o materials d'investigació i docència en els termes establerts a l'art. 32 del Text Refós de la Llei de Propietat Intel·lectual (RDL 1/1996). Per altres utilitzacions es requereix l'autorització prèvia i expressa de la persona autora. En qualsevol cas, en la utilització dels seus continguts caldrà indicar de forma clara el nom i cognoms de la persona autora i el títol de la tesi doctoral. No s'autoritza la seva reproducció o altres formes d'explotació efectuades amb finalitats de lucre ni la seva comunicació pública des d'un lloc aliè al servei TDX. Tampoc s'autoritza la presentació del seu contingut en una finestra o marc aliè a TDX (framing). Aquesta reserva de drets afecta tant als continguts de la tesi com als seus resums i índexs.
dc.sourceTDX (Tesis Doctorals en Xarxa)
dc.subjectanàlisis global de Darboux
dc.subjectsolucions antisimètriques
dc.subjectestructuras de Poisson
dc.subjectequacions diferencials parcials
dc.subjectequacions de Jacobi
dc.titleThe Jacobi identities for finite-dimensional Poisson structures: a P.D.E. based analysis of some new constructive results and solution families

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