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论文范文
1. Introduction Compared with the integer order model, the fractional order model can describe the mechanical and physical behavior of the complex system more accurately. Miller and Ross [1] once pointed out that almost every field of science and engineering involves fractional calculus. Fractional calculus has been widely used in mathematics, physics, chemistry, signal processing, engineering, and so on [2]. Riemann-Liouville fractional derivative and Caputo fractional derivative are in common use. Cresson [3] presented two more general fractional derivatives in 2006, i.e., the combined Riemann-Liouville fractional derivative and the combined Caputo fractional derivative. The study of the fractional calculus of variations was started in 1996, when Riewe [4, 5] was considering how to deal with the friction force and other forms of dissipative force in the classical mechanics and the quantum mechanics. Since then, Agrawal [6, 7], Atanacković [8], Almeida [9, 10], and some other scholars [11, 12] also studied fractional variational problems. However, all of the results were related to Lagrangian system or Hamiltonian system. As a matter of fact, there is a more general system called Birkhoffian system, which was introduced in 1927 [13]. Birkhoffian mechanics is a generalization of Hamiltonian mechanics, which is described in detail in [14]. Birkhoffian dynamics has gained significant headways [15, 16]. Recently, in [17], Luo first established Birkhoffian mechanics with the combined Riemann-Liouville fractional derivative and the combined Caputo fractional derivative. There is a set of unique integral theory in analytical mechanics, which is useful in solving differential equations of motion. Besides, symmetry and conserved quantity can help to reveal the intrinsic physical properties of the dynamic system. The commonly used symmetry methods [18] are Noether symmetry method, Lie symmetry method, and Mei symmetry method. Noether symmetry refers to the invariance of the Hamilton action under the infinitesimal transformations [19]. Lie symmetry refers to the invariance of the differential equations of motion under the infinitesimal transformations [20]. Mei symmetry means the invariance of the forms of the differential equations of motion when the dynamical functions, such as the kinetic energy, the potential energy, the generalized forces, the Lagrangian, the Hamiltonian, and the Birkhoffian, are replaced by the transformed functions under the infinitesimal transformations. Many important research results have been achieved in terms of symmetry and conserved quantity of the constrained mechanical systems [21]. The fractional symmetry and conserved quantity were first studied by Frederico and Torres [22]. Based on the Riemann-Liouville fractional derivative, they established the fractional Noether theorem. Bourdin et al. [23] presented a new expression of the fractional conserved quantity and took the fractional harmonic oscillator as an example. Based on the Caputo fractional derivative, Muslih [24] extended the fractional Noether theorem of finite degree of freedom to the fractional field theory and presented the conserved quantity of the fractional Dirac field. In recent years, Zhang [25, 26], Jia [27], and Zhang [28] studied the Noether symmetry and conserved quantity of the fractional Birkhoffian system. Fu [29] investigated the Lie symmetry of the fractional nonholonomic Hamiltonian system and the corresponding inverse problem. Luo [30] presented the Mei symmetry and conserved quantity of the generalized fractional Hamiltonian system. 
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