Sasakian Structure Associated with a Second-Order ODE and Hamiltonian Dynamical Systems
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Date
2022
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Springer Singapore Pte Ltd
Access Rights
info:eu-repo/semantics/openAccess
Abstract
We define a contact metric structure on the manifold corresponding to a second-order ordinary differential equation d(2)y/dx(2) = f (x, y, y') and show that the contact metric structure is Sasakian if and only if the 1-form 1/2 (dp - fdx) defines a Poisson structure. We consider a Hamiltonian dynamical system defined by this Poisson structure and show that the Hamiltonian vector field, which is a multiple of the Reeb vector field, admits a compatible bi-Hamiltonian structure for which f can be regarded as Hamiltonian function. As a particular case, we give the compatible bi-Hamiltonian structure of the Reeb vector field such that the structure equations correspond to the Maurer-Cartan equations of an invariant coframe on the Heisenberg group and the independent variable plays the role of Hamiltonian function. We also show that the first Chern class of the normal bundle of an integral curve of a multiple of the Reeb vector field vanishes iff f(x) + ff(p) = Psi (x) for some Psi.
Description
Keywords
Sasakian Structure, Poisson Structure, Hamiltonian Systems, Ordinary Differential Equations, Thermodynamic Phase-Space, Contact Metric Manifolds, Geometry
Journal or Series
Bulletin Of The Iranian Mathematical Society
WoS Q Value
Q3
Scopus Q Value
Q3
Volume
48
Issue
4
