A general method of fractional dynamics, i.e., fractional Jacobi last multiplier method, and its applications

In this paper, we present a general method of finding conserved quantities, i.e., fractional Jacobi last multiplier method, and explore its applications to fractional dynamical system. Based on the definition of Riesz-Riemann-Liouville fractional derivative, we study general fractional dynamical equations, construct its fractional Jacobi last multiplier, and, respectively, give the determining equation and three important properties of the multiplier. And then, we present the fractional Jacobi last multiplier method, which includes three theorems of finding conserved quantities of fractional dynamical systems. Further, in fractional framework, we explore the relation between Lie symmetry and Jacobi last multiplier. Furthermore, the fractional Jacobi last multiplier method is applied to the fractional generalized Hamiltonian system, the complete fractional Hamiltonian system, the standard fractional Hamiltonian system, the fractional Nambu system and the fractional Birkhoffian system, and five corresponding propositions are given. Also by using the fractional Jacobi last multiplier method, we respectively find the conserved quantities of a fractional relativistic Buchdahl model, a fractional Euler-Poinsot model and a fractional Duffing oscillator model. This work constructs a basic theoretical framework of fractional Jacobi last multiplier method, and provides a general method of fractional dynamics.