Fractional Birkhoffian mechanics

Abstract

In this paper, we present a new fractional dynamical theory, i.e., the dynamics of a Birkhoffian system with fractional derivatives (the fractional Birkhoffian mechanics), which gives a general method for constructing a fractional dynamical model of the actual problem. By using the definition of combined fractional derivative, we present a unified fractional Pfaff action and a unified fractional Pfaff-Birkhoff principle, and give four kinds of fractional Pfaff-Birkhoff principles under the different definitions of fractional derivative. And then, by using the fractional Pfaff-Birkhoff principles, we establish a series of fractional Birkhoffian equations with different fractional derivatives and construct the tensor representation of autonomous fractional Birkhoffian equations. Further, we study the relationship among the fractional Bikhoffian system, the fractional Hamiltonian system and the fractional Lagrangian system and give the transformation conditions. And furthermore, as applications of the fractional Birkhoffian method, we construct five kinds of fractional dynamical models, which include the fractional Lotka biochemical oscillator model, the fractional Whittaker model, the fractional Hojman-Urrutia model, the fractional Hénon-Heiles model and the fractional Emden model.

Fractional Nambu dynamics

“…(1)- (6)]. Based on the fractional operators (1)-(6), using variational calculus, scientists could construct a series of fractional dynamical equations in physics and engineering; however, because of difficulty of integration by parts with fractional derivative, the researchers had to use the transversality condition in calculations, which led z − 1 ≤ α, β < z reduce to 0 ≤ α, β < 1 [38][39][40][41][42]53]. So, for actual fractional dynamical systems, researchers have attained many novel and interesting results that are physically plausible in 0 ≤ α, β < 1.…”

“…Equation (41) has been given in Ref. [53]. Equation (39) is a new kind of autonomous fractional Birkhoffian equations, which is more general than Eqs.…”

“…Since then, the study of the basic theories and methods for fractional dynamics has become a hot topic, and various types of dynamical systems with fractional derivatives have been widely studied in the literature, such as fractional Lagrangian mechanics, fractional Hamiltonian mechanics and fractional nonholonomic mechanics [35][36][37][38][39][40][41][42][43][44][45][46][47]. Recently, we established fractional generalized Hamiltonian mechanics and fractional Birkhoffian mechanics and constructed their basic theoretical framework, respectively [48][49][50][51][52][53][54]. The basic theories and methods of fractional dynamics have been extensively applied in many fields of science and engineering.…”

A new method of dynamical stability, i.e. fractional generalized Hamiltonian method, and its applications

“…This important discovery caused the shock of science, and scientists began to study many problems about the dynamical system with fractional derivatives. Since then, the study of the basic theories and methods for fractional dynamics has become a hot topic, and won wide development in theories and applications [26][27][28][29][30][31][32][33][34][35][36][37][38][39]. Recently, we established the fractional generalized Hamiltonian mechanics, which include its gradient representation, Lie algebraic structure, generalized Poisson conservation law, variation equations, construction method of integral invariants, and so on [40][41][42][43].…”

Noether theorem for non-conservative systems with time delay in phase space based on fractional model

“…Atanacković et al [23] studied the fractional Noether theorem within the RiemannLiouvill fractional derivatives based on the concept of classical conserved quantity. Zhang and his coworkers [24][25][26][27][28] and Luo [29] studied the fractional Birkhoffian mechanics and its Noether symmetries. Fu and Zhou [30,31] studied the equations of fractional nonholonomic system and the symmetry theories of fractional Hamilton system.…”