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We define a contact metric structure on the manifold corresponding to asecond order ordinary differential equation $d^2y/dx^2=f(x,y,y')$ and show thatthe contact metric structure is Sasakian if and only if the 1-form$\frac{1}{2}(dp-fdx)$ defines a Poisson structure. We consider a Hamiltoniandynamical system defined by this Poisson structure and show that theHamiltonian vector field, which is a multiple of the Reeb vector field, admitsa compatible bi-Hamiltonian structure for which $f$ can be regarded as aHamiltonian function. As a particular case, we give a compatible bi-Hamiltonianstructure of the Reeb vector field such that the structure equations correspondto the Maurer-Cartan equations of an invariant coframe on the Heisenberg groupand the independent variable plays the role of a Hamiltonian function. We alsoshow that the first Chern class of the normal bundle of an integral curve of amultiple of the Reeb vector field vanishes iff $f_x+ff_p = \Psi (x)$ for some$\Psi$.

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