We show that all of the curve motions specified in the Frenet-Serret frame are described by the time evolution of an integral curve of a time-like Hamiltonian dynamical system in Minkowski space such that the integral curve under consideration is a geodesic curve on a leaf of the foliation determined by the Poisson structure. Accordingly, any nonlinear soliton equation related to curve dynamics is obtained as the time evolution of an integral curve of a Hamiltonian system. As an expository example, we define Hashimoto function in the Darboux frame which is reduced to the classical Hashimoto function provided that the Poisson vector corresponds to principal normal of an integral curve and show that the defocusing version of the nonlinear Schrödinger equation and the mKdV equation are obtained by the time evolution of this function.