Fractional time-scales Noether theorem with Caputo derivatives for Hamiltonian systems
Introduction
In 1918, Noether [1], a German female mathematician, published an influential paper in German called “invariant variational problems”. Later, this article was translated into English [2] so that more people could read and study it. The paper mainly gave two theorems: the first theorem concerns symmetries and conserved quantities in classical mechanics, and the second theorem involves in general relativity. The first theorem of Noether (Noether theorem) has become the basis of symmetries and conserved quantities not only for the study of classical mechanics and classical field theory but also for the study of quantum mechanics and quantum field theory. Not only can Noether theorem help to find the classical conservation law of Newtonian mechanics, the generalized momentum conservation law and generalized energy conservation law of Lagrangian mechanics, but also find more conservation laws. The conservation laws of mechanical systems are of great significance to the study of dynamic behavior, stability and calculation of mechanical systems [3]. Therefore, Noether theorem plays a crucial role in analytical mechanics. With the hot research on Noether theorem in recent decades, the research on Noether theorem of classical mechanics [4], [5], Lagrangian mechanics [6], [7], [8], Hamiltonian mechanics [6], [7], [9], nonholonomic mechanics [10], [11] and Birkhoffian mechanics [6], [7] has been becoming more and more perfect. Not just for these continuous systems, but Noether theorems for discrete systems have obtained some results [12], [13], [14]. In recent years, some scholars have used the time-scales theory to unify the study of Noether theorem for continuous systems and discrete systems.
The concept of time-scales is a unified theory of continuous and discrete calculus proposed by Hilger in 1988 [15]. He found a deep connection between continuous calculus and difference equations. Instead of being two separate theories, they were really two strands of the same thing. Accordingly, Hilger devised the time-scales theory to summarize them. It is paramount to note that not only can the time-scales theory unify the study of continuous and discrete systems, but also it can study quantum systems simultaneously. There is no doubt that this advantageous mathematical tool has brought a lot of conveniences to build models and has been applied in economics, biology and other disciplines [16], [17], [18]. As for the time-scales Noether theorem, it dates back to 2008 when Bartosiewicz and Torres studied the time-scales Noether theorem with derivatives for Lagrangian systems by using the technique of time-re-parameterization [19]. Then, in 2011, Bartosiewicz et al. established the time-scales second Euler-Lagrange equation with derivatives [20]. Subsequently, time-scales Noether theorems were extended to Hamiltonian mechanics [21], [22], [23], [24], nonconservative mechanics [25], [26], [27], nonholonomic mechanics [28], [29], [30], Birkhoffian mechanics [31], [32], [33] and so on. However, in 2016, Anerot et al. pointed out the error of Bartosiewicz [19], [20] and reproved the time-scales Noether theorem by using the generalized Jost’s method [34], [35]. Consequently, the time-scales Noether theorems obtained by referring to the technique of time-re-parameterization in [19] and [20] are open to question.
Due to the rapid development of time-scales calculus and the need for more practical mathematical models, the emergence of fractional time-scales calculus was promoted. As is known to all, in mathematical concept, fractional calculus is a natural extension of integral calculus, and in practice, it is found that fractional damping does exist in nature and practical problems. Compared with integral calculus, fractional model can provide more practical models for engineering, materials and other fields. In 1999, Podlubny [36] gave the definitions of fractional derivatives for continuous systems. If the time-scale satisfies then the fractional time-scale calculus is consistent with the classical fractional calculus in the monograph [36]. In 2007, Atici and Eloe [37] proposed the definition of discrete fractional derivatives and finite fractional differences. If the time-scale is taken to be the fractional time-scale calculus becomes the calculus of the discrete system with step size in [37]. If the time-scale is the fractional time-scale calculus becomes the discrete fractional -calculus in [38]. Therefore, continuous, discrete and quantum fractional systems can be processed systematically with different values of time-scales . Based on the fractional calculus on time-scales, the similarities and differences between continuous and discrete fractional systems can be analyzed, which is beneficial to avoid the repeated proof of some problems. In addition, the step size can not only be a constant but also become a function with variables. Based on the different choices of time-scales, more and more general results can be obtained, which can describe the physical essences of continuous and discrete fractional systems as well as other complex fractional dynamic systems more clearly and accurately. Because of these advantages of fractional time-scales calculus, in recent years some results have been obtained, including fractional time-scales principles and inequalities [39], existence and uniqueness of solutions for fractional time-scales dynamic equations [40], [41], the theory of fractional time-scales calculus [42], [43], [44], fractional time-scales operators with application to dynamic equations [45], [46], fractional time-scales optimal control [47], fractional time-scales chaotic systems [48], [49], [50], fractional time-scales recurrent neural networks [51]. Although fractional Noether theorems for Lagrangian systems [52], Birkhoffian systems [53], [54], [55] and nonconservative systems [56], [57], [58] have been studied, there are no reports on fractional time-scales Noether theorem so far.
In this paper, the fractional time-scales Noether theorem with Caputo derivatives for Hamiltonian systems is investigated. Compared with Lagrangian systems, Hamiltonian systems are symmetrical and symplectic in structure, which led to the development of geometric mechanics. In addition, in solving many complex mechanical problems, such as celestial mechanics and vibration theory, it is more convenient to have a general discussion in Hamiltonian systems. Moreover, under certain conditions, the fractional time-scales Noether theorem for Hamiltonian systems can reduce to the one for Lagrangian systems. This paper proceeds as follows. In Section 2, some basics and fundamental properties of fractional time-scales calculus are recalled. In Section 3, the fractional time-scales Hamilton canonical equations with Caputo derivatives are formulated. The time-scales Noether theorems with Caputo derivatives for Hamiltonian systems without transforming time and with transforming time are obtained in Section 4 and Section 5, respectively. In the next section, two examples are presented. Section 7 summarizes the results of this investigation.
Section snippets
Preliminaries
The definitions of time-scales are given in detail in the books [16], [18], [59], including forward jump operator backward jump operator graininess function derivative and the set of rd-continuous functions . In this section, some main definitions and properties of fractional time-scales derivatives are recalled. Definition 1 The rd-continuous functions are defined bywhere . So, it is easy to get that[59]
Fractional time-scales Hamiltonian canonical equations with Caputo derivatives
Assume the configuration of a time-scales mechanical system is determined by generalized coordinates and the time-scales Lagrangian of the system with Caputo derivatives is . Then, the generalized momenta areand its Hamiltonian isSo the Hamiltonian action with Caputo derivatives is
The fractional time-scales Hamilton
Fractional time-scales Noether theorem for Hamiltonian systems without transforming time
Introduce the infinitesimal transformations of a one-parameter group without transforming timewhere are the infinitesimal generators and is an infinitesimal parameter. Definition 7 The Hamiltonian action (10) is invariant under the infinitesimal transformations (26) iffor any subinterval with . Criterion 1 If the Hamiltonian
Fractional time-scales Noether theorem for Hamiltonian systems with transforming time
Next, introduce the infinitesimal transformations of the -admissible projectable group [35] with an infinitesimal parameter in the case of transforming timewhere are the infinitesimal generators. Definition 8 The Hamiltonian action (10) is invariant under a one-parameter -admissible projectable group of transformations (34) if and only if
Examples
Example 1 Because fractional models have a unique ability in describing anomalous behavior and memory effects, fractional dynamic equations lead to better results than integer equations in many practical systems. The fractional damped oscillator is a generalization of the classical damped oscillator equation, which can be considered to describe the dynamics of certain gases dissolved in a fluid and the dynamics of a sphere immersed in an incompressible viscous fluid [61]. The time-scales theory is
Conclusions
The fractional time-scales Noether theorems without transforming time (Theorem 1) and with transforming time (Theorem 2) for Hamiltonian systems are studied. In fact, Theorem 1 can become the results of literature [19], [22], [31] under certain conditions. Since the time-scales Jost’s method [35] can deal with time-scales Noether theorem and the fractional Jost’s method [52] can work out fractional Noether theorem, the fractional time-scales Noether theorem can be solved by combining these two
Declaration of Competing Interest
The authors declare that they have no conflict of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (grant number 11972241); and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (grant number KYCX20_0251).
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