Sasakian structure associated with a second order ODE and Hamiltonian dynamical systems

Abstract

We show that the existence of a Sasakian structure on a manifold corresponding to a second order ordinary differential equation (ODE) is equivalent to the existence of the Poisson structure determined by a one-form. We consider Hamiltonian dynamical system associated with this Poisson structure and show that the compatibility condition for the bi-Hamiltonian structure of the Reeb vector field manifests that the structure equations for the coframe encoding a certain family of second order ODEs are the Maurer-Cartan equations for the Heisenberg group and the independent variable plays the role of a Hamiltonian function.