Sasakian Structure Associated with a Second-Order ODE and Hamiltonian Dynamical Systems

Abstract

We define a contact metric structure on the manifold corresponding to a second order ordinary differential equation d2y/dx2 = f(x, y, y) and show that the contact metric structure is Sasakian if and only if the 1-form 12(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 a Hamiltonian function. As a particular case, we give a 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 a 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 fx + ffp = Ψ(x) for some Ψ.