TY - BOOK

T1 - Singularities of Poisson structures and Hamiltonian bifurcations

AU - Meer, van der, J.C.

PY - 2010

Y1 - 2010

N2 - 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).

AB - 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).

M3 - Report

T3 - CASA-report

BT - Singularities of Poisson structures and Hamiltonian bifurcations

PB - Technische Universiteit Eindhoven

CY - Eindhoven

ER -