A New Fractional Gradient Representation of Birkhoff Systems

A fractional generalization of the gradient system to the Birkhoff mechanics, that is, a new fractional gradient representation of the Birkhoff system is investigated, in this paper. The definitions of the fractional gradient system are generalized to Birkhoff mechanics. Based on the definition, a general condition that a Birkhoff system can be a fractional gradient system is derived, the former studies for fractional gradient representation of the Birkhoff system are special cases of this paper. As applications of the results, the Birkhoff equations and fractional gradient expression of several classical nonlinear models are derived, such as the Hénon–Heiles equation, the Duffing oscillator model, and the Hojman–Urrutia equations. The results indicate, different from the former studies, that only gave the second-order gradient (integer order) expression for the Birkhoff system, an arbitrary fractional order gradient representation, exists for the Birkhoff system. The fractional potential function obtained from the general condition can determine the stationary states of these models in Birkhoffian expression.