Fractional Nambu dynamics

Abstract

In this paper, we present a new fractional dynamical theory, i.e., the dynamics of a Nambu system with fractional derivatives (the fractional Nambu dynamics), and study its applications to mechanical systems. By using the definition of combined fractional derivative, we construct unified fractional Nambu equations and, respectively, give four new kinds of fractional Nambu equations based on the different definitions of fractional derivative. Further, we study the relations between the fractional Nambu system, the fractional generalized Hamiltonian system, the fractional Birkhoffian system, the fractional Hamiltonian system, the fractional Lagrangian system and a series of integer-order dynamical systems and present the transformation conditions. And furthermore, as applications of the fractional Nambu method, we construct two kinds of fractional dynamical models, which include the fractional Euler–Poinsot model of rigid body which rotates with respect to a fixed point and the fractional Yamaleev oscillator model. This work provides a general method for studying a fractional dynamical problem which is related to science and engineering.