Hamiltonian dynamical systems and geometry of surfaces in 3-D

Abstract Hamiltonian vector field, Poisson vector field and the gradient of Hamiltonian function defines Darboux frame along an integral curve of a Hamiltonian dynamical system on a surface whose normal vector field corresponds to the Poisson structure for a given Hamiltonian system. We show that the existence of compatible Poisson structures determined by the normal legs of the Darboux frame is resolved to the characteristic equation for the Weingarten map. We also show that a Hamiltonian dynamical system in three dimensions has bi-Hamiltonian representation determined by the normal legs of Frenet-Serret triad if and only if an integral curve of Hamiltonian vector field is both a geodesic and a line of curvature.