  Working bi-Hamiltonian structure and Jacobi identity in Frenet-Serret frame
associated to a dynamical system, we proved that all dynamical systems in three
dimensions possess two compatible Poisson structures. We investigate relations
between geometry of surfaces defined by potential function of a gradient system
and its bi-Hamiltonian structure. We show that it is possible to find
Hamiltonian functions whose gradient flows have geodesic curvature zero on
potential surfaces. Using this, we conclude that Hamiltonian functions are
determined by distance functions on potential surfaces. We apply this technique
to find conserved quantities of three dimensional gradient systems including
the Aristotelian model of the three-body motion.
