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 reprametrizations 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.

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Cited by in corpus (10)

• Generalized results on the role of new-time transformations in finite-dimensional Poisson systems
• Periodic orbits in analytically perturbed Poisson systems
• Inverse Jacobi multiplier as a link between conservative systems and Poisson structures
• New global solutions of the Jacobi partial differential equations
• Generalization of the separation of variables in the Jacobi identities for finite-dimensional Poisson systems
• A stability criterion for non-degenerate equilibrium states of completely integrable systems
• Poisson systems as the natural framework for additional first integrals via Darboux invariant hypersurfaces
• Bifurcations of limit cycles of perturbed completely integrable systems
• Congruence method for global Darboux reduction of finite-dimensional Poisson systems
• Poisson brackets after Jacobi and Plucker