Lie algebraic structure and generalized Poisson conservation law for fractional generalized Hamiltonian systems

Abstract

Based on definition of Riemann–Liouville fractional derivatives, we find that the fractional generalized Hamiltonian system possesses consistent algebraic structure and Lie algebraic structure, and the generalized Poisson conservation law of the fractional generalized Hamiltonian system is investigated. As special cases of this paper, the conditions under which a fractional generalized Hamiltonian system can be reduced to a generalized Hamiltonian system and a classical Hamiltonian system are given, and the Lie algebraic structure and Poisson conservation law of these special dynamical systems are obtained, respectively. Then, by using the method and results of this paper, we construct three kinds of new fractional dynamical model, i.e. a fractional Henon–Heiles model, a fractional Euler–Poinsot model of a rigid body and a fractional Volterra model of the three species groups, and their Poisson conserved quantities are obtained.