Fractional differential equations of motion in terms of combined Riemann—Liouville derivatives

Abstract

In this paper, we focus on studying the fractional variational principle and the differential equations of motion for a fractional mechanical system. A combined Riemann—Liouville fractional derivative operator is defined, and a fractional Hamilton principle under this definition is established. The fractional Lagrange equations and the fractional Hamilton canonical equations are derived from the fractional Hamilton principle. A number of special cases are given, showing the universality of our conclusions. At the end of the paper, an example is given to illustrate the application of the results.

Motion equations and non-Noether symmetries of Lagrangian systems with conformable fractional derivative

“…We present the definitions and properties of conformable fractional calculus [19,20] to benefit the readers.…”

“…The symmetries, conservation laws and bifurcation at al of non-linear differential equations in mathematical physics have been paid much attentions [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Lately, Cai et al [6] has studied the non-Noether conserved quantities for holonomic mechanical system.…”

“…Agrawal [19], presented the fractional variational problems with left and right Riemann-Liouville derivatives. Zhang [20], further gave the fractional differential equations in terms of combined Riemann-Liouville derivatives. However, the definitions of these fractional order derivatives are mostly in the form of integrals, which bring great difficulties for general researchers and many engineering and technical personnel.…”

Fractional Birkhoffian Mechanics Based on Quasi-Fractional Dynamics Models and Its Noether Symmetry

“…Fractional calculus is an important mathematical tool in science and engineering [25][26][27][28]. In recent decades, the research of fractional calculus has developed greatly, and its application fields have expanded to automatic control, quantum mechanics, and mechanical systems [29][30][31][32][33][34][35]. Riewe [36,37] introduced the fractional variational problem for the first time in the study of nonconservative mechanics.…”

Fractional Birkhoffian mechanics

“…In the end of the 1970s, Mandelbrot [26] discovered that a large number of fractional dimension examples exists in nature. Since then, the study of the fractional dynamics has become a hot topic, and won wide development in theories and applications, including fractional Lagrangian mechanics, fractional Hamiltonian mechanics, fractional dynamics of nonholonomic system and fractional generalized Hamiltonian mechanics [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43], and their application [44][45][46][47][48][49][50]. In order to better solve the fractional dimension problems in science and engineering, it is necessary to propose a fractional dynamical theory of Birkhoffian systems.…”