Singularities of Poisson structures and Hamiltonian bifurcations

Research output: Book/ReportReportAcademic

31 Downloads (Pure)

Abstract

Consider a Poisson structure on C8(R3,R) with bracket {, } and suppose that C is a Casimir function. Then {f, g} =<¿C, (¿g x ¿f) > is a possible Poisson structure. This confirms earlier observations concerning the Poisson structure for Hamiltonian systems that are reduced to a one degree of freedom system and generalizes the Lie-Poisson structure on the dual of a Lie algebra and the KKS-symplectic form. The fact that the governing reduced Poisson structure is described by one function makes it possible to find a representation, called the energy-momentum representation of the Poisson structure, describing both the singularity of the Poisson structure and the singularity of the energy-momentum mapping and hence the bifurcation of relative equilibria for such systems. It is shown that Hamiltonian Hopf bifurcations are directly related to singularities of Poisson structures of type sl(2).
Original languageEnglish
Place of PublicationEindhoven
PublisherTechnische Universiteit Eindhoven
Number of pages15
Publication statusPublished - 2010

Publication series

NameCASA-report
Volume1005
ISSN (Print)0926-4507

Fingerprint

Dive into the research topics of 'Singularities of Poisson structures and Hamiltonian bifurcations'. Together they form a unique fingerprint.

Cite this