Fractional Birkhoffian System Research Articles

Generalization of Mei symmetry approach to fractional Birkhoffian mechanics

The fractional Birkhoffian systems are investigated for Mei symmetry and corresponding conserved quantities. It is divided into two cases, one is standard fractional Birkhoffian systems, the other is fractional Birkhoffian systems under quasi-fractional dynamics models of El-Nabulsi type. The fractional Pfaff-Birkhoff principles are presented, and Birkhoff’s equations are deduced. Mei symmetry and its determining equations are established. Mei symmetry theorems are proved and fractional conserved quantities are obtained.

A general numerical scheme for the optimal control of fractional Birkhoffian systems

This paper gives a general numerical scheme for the optimal control problem of fractional Birkhoffian systems. The fractional forced Birkhoff equations within Riemann–Liouville fractional derivatives are derived from the fractional Pfaff–Birkhoff–d’Alembert principle which includes the control as an external force term. Following the strategy of variational integrators, the fractional Pfaff–Birkhoff–d’Alembert principle is directly discretized to develop the equivalent discrete fractional forced Birkhoff equations that served as the equality constraints of the optimization problem. Together with the initial and final state constraints on the configuration space, the original optimal control problem is converted into a nonlinear optimization problem subjected to a system of algebraic constraints, which can be solved by existing algorithms. An illustrative example is given to show the efficiency and simplicity of the proposed method.

Noether’s theorem for fractional Birkhoffian system of Herglotz type with time delay

Abstract Noether symmetry theorem of Herglotz type for time-delayed fractional Birkhoffian system are studied. Firstly, based on the fractional derivative of Riemann-Liouville, the Herglotz variational principle of time-delayed fractional Birkhoffian system is established, and the time-delayed Birkhoff′s equation of Herglotz type is derived. Secondly, the definition and criterion of Herglotz type Noether symmetric transformation of time-delayed fractional Birkhoffian system are established. Thirdly, the Noether′s theorem of the system is proposed and proved, in addition, the inner relationship between Noether symmetries and conservation is accurately explored. Next, the special case of the theorem is discussed, in other words, when the Herglotz generalized variational principle is reduced to the classical variational principle, the result of this paper is degraded into the Noether symmetry theorem of the time-delayed fractional Birkhoffian system. Finally, an example is given.

Conserved quantities and adiabatic invariants of fractional Birkhoffian system of Herglotz type**Project supported by the National Natural Science Foundation of China (Grant Nos. 11972241, 11572212, and 11272227), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20191454), and the Innovation Program for Postgraduade in Higher Education Institutions of Jiangsu Province, China (Grant No. KYCX19 2013).

In order to further study the dynamical behavior of nonconservative systems, we study the conserved quantities and the adiabatic invariants of fractional Brikhoffian systems with four kinds of different fractional derivatives based on Herglotz differential variational principle. Firstly, the conserved quantities of Herglotz type for the fractional Brikhoffian systems based on Riemann–Liouville derivatives and their existence conditions are established by using the fractional Pfaff–Birkhoff–d′Alembert principle of Herglotz type. Secondly, the effects of small perturbations on fractional Birkhoffian systems are studied, the conditions for the existence of adiabatic invariants for the Birkhoffian systems of Herglotz type based on Riemann–Liouville derivatives are established, and the adiabatic invariants of Herglotz type are obtained. Thirdly, the conserved quantities and adiabatic invariants for the fractional Birkhoffian systems of Herglotz type under other three kinds of fractional derivatives are established, namely Caputo derivative, Riesz–Riemann–Liouville derivative and Riesz–Caputo derivative. Finally, an example is given to illustrate the application of the results.

Perturbation to Noether symmetry for fractional dynamic systems of variable order

Fractional dynamic system of variable order is one of the interesting topics to study. Perturbation to symmetry, which is closely related to its integrability, is worth to be well studied. In this paper, Perturbation to Noether symmetry and adiabatic invariant are investigated in terms of Riemann–Liouville fractional derivative of variable order for fractional generalized Birkhoffian system. Then under some special conditions and transformations, adiabatic invariants for the fractional Birkhoffian system, the fractional Hamiltonian system and the fractional Lagrangian system are discussed. It has been observed that some results obtained from special cases are new and others are consistent with the existing results, which reconfirm the credibility of proposed achievements. Finally, an example is given to illustrate the methods and results.

Adiabatic Invariants for Generalized Fractional Birkhoffian Mechanics and Their Applications

Perturbation to Noether symmetry and adiabatic invariants are investigated for the generalized fractional Birkhoffian system with the combined Riemann-Liouville fractional derivative and the combined Caputo fractional derivative, respectively. Firstly, differential equations of motion for the generalized fractional Birkhoffian system are established. Secondly, Noether symmetry and conserved quantity are studied. Thirdly, perturbation to Noether symmetry and adiabatic invariants are presented for the generalized fractional Birkhoffian mechanics. And finally, several applications are discussed to illustrate the results and methods.

Noether Symmetry and Conserved Quantities of Fractional Birkhoffian System in Terms of Herglotz Variational Problem**Supported by the National Natural Science Foundation of China under Grant Nos. 11272227 and 11572212, the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province (KYZZ16 0479), and the Innovation Program for Postgraduate of Suzhou University of Science and Technology (SKCX16 058)

The aim of this paper is to study the Herglotz variational principle of the fractional Birkhoffian system and its Noether symmetry and conserved quantities. First, the fractional Pfaff-Herglotz action and the fractional Pfaff-Herglotz principle are presented. Second, based on different definitions of fractional derivatives, four kinds of fractional Birkhoff’s equations in terms of the Herglotz variational principle are established. Further, the definition and criterion of Noether symmetry of the fractional Birkhoffian system in terms of the Herglotz variational problem are given. According to the relationship between the symmetry and the conserved quantities, the Noether’s theorems within four different fractional derivatives are derived, which can reduce to the Noether’s theorem of the Birkhoffian system in terms of the Herglotz variational principle under the classical conditions. As applications of the Noether’s t heorems of the fractional Birkhoffian system in terms of the Herglotz variational principle, an example is given at the end of this paper.

Perturbation Theory of Fractional Lagrangian System and Fractional Birkhoffian System

Perturbation to symmetry and adiabatic invariants are studied for the fractional Lagrangian system and the fractional Birkhoffian system in the sense of Riemann-Liouville derivatives. Firstly, the fractional Euler-Lagrange equation, the fractional Birkhoff equations as well as the fractional conservation laws for the two systems are listed. Secondly, the definition of adiabatic invariant for fractional mechanical system is given, then perturbation to symmetry and adiabatic invariants are established for the fractional Lagrangian system and the fractional Birkhoffian system under the special and general infinitesimal transformations, respectively. Finally, two examples are devoted to illustrate the results.

Noether Symmetry and Conserved Quantity for FractiOnal Birkhoffian Mechanics and Its Applications

AbstractNoether theorem is an important aspect to study in dynamical systems. Noether symmetry and conserved quantity for the fractional Birkhoffian system are investigated. Firstly, fractional Pfaff actions and fractional Birkhoff equations in terms of combined Riemann-Liouville derivative, Riesz-Riemann-Liouville derivative, combined Caputo derivative and Riesz-Caputo derivative are reviewed. Secondly, the criteria of Noether symmetry within combined Riemann-Liouville derivative, Riesz-Riemann-Liouville derivative, combined Caputo derivative and Riesz-Caputo derivative are presented for the fractional Birkhoffian system, respectively. Thirdly, four corresponding conserved quantities are obtained. The classical Noether identity and conserved quantity are special cases of this paper. Finally, four fractional models, such as the fractional Whittaker model, the fractional Lotka biochemical oscillator model, the fractional Hénon-Heiles model and the fractional Hojman-Urrutia model are discussed as examples to illustrate the results.

Fractional symmetrical perturbation method of finding adiabatic invariants of disturbed dynamical systems

For a disturbed dynamical system that can be transformed into fractional Birkhoffian representation with all dynamical information, under a more general fractional infinitesimal transformation of Lie group with high time extension and fractional extension, a new kind of fractional Mei symmetrical perturbation method which is most universal significance is presented and it is found that, using the new method, we can find a new kind of non-Noether adiabatic invariant; as the special cases of new method, we, respectively, reveal an autonomous disturbed fractional Birkhoffian system possesses more adiabatic invariants, a new kind of non-Noether exact invariant directly led by fractional Mei symmetry and a new kind of non-Noether exact and adiabatic invariant of integer Birkhoffian systems. Also, as the new method’s applications to nonlinear dynamical problems, we, respectively, explore the symmetrical perturbation and adiabatic invariant of a disturbed fractional general relativistic Buchdahl model and a disturbed fractional Emden model. It is worth pointing out that, for a disturbed dynamical system, this work reveals intrinsic relation between the fractional symmetrical perturbation and the adiabatic invariant, and provides a general method for finding adiabatic invariants of an actual disturbed fractional dynamical system that is related to science and engineering.

Fractional conformal invariance method for finding conserved quantities of dynamical systems

For a dynamical system that can be transformed into fractional Birkhoffian representation, under a more general kind of fractional infinitesimal transformation of Lie group, we present the fractional conformal invariance method and it is found that, using the new method, we can find a new kind of non-Noether conserved quantity; and we find that, as a special case, an autonomous fractional Birkhoffian system possesses more conserved quantities. Also, as the fractional conformal invariance method’s applications, we, respectively, explore the conformal invariance and conserved quantities of a fractional Lotka biochemical oscillator and a fractional Hojman–Urrutia model. This work constructs a basic theoretical framework of fractional conformal invariance method, and provides a general method for finding conserved quantities of an actual fractional dynamical system that is related to science and engineering.

Noether's theorem of fractional Birkhoffian systems

In this paper, we study Noether type symmetry theorem to fractional Birkhoffian systems with Riemann–Liouville derivatives. This theorem provides an explicit algorithmic way to compute a constant for any Birkhoffian systems admitting a symmetry. Finally, we extend our Noether's theorem to fractional Birkhoffian systems base on Caputo or Riesz derivatives.

Variational integrators for fractional Birkhoffian systems

In this paper, we generalize the Pfaff–Birkhoff principle to the case of containing fractional derivatives and obtain the so-called fractional Pfaff–Birkhoff–d’Alembert principle. The fractional Birkhoff equations in the sense of Riemann–Liouville fractional derivative are derived. Under the framework of variational integrators, we develop the discrete fractional Birkhoff equations by approximating the Riemann–Liouville fractional derivative with the shifted Grunwald–Letnikov fractional derivative. The resulting algebraic equations can be served as an algorithm to numerically solve the fractional Birkhoff equations. A numerical example is demonstrated to show the validity and applicability of the presented methodology.

A general method of fractional dynamics, i.e., fractional Jacobi last multiplier method, and its applications

In this paper, we present a general method of finding conserved quantities, i.e., fractional Jacobi last multiplier method, and explore its applications to fractional dynamical system. Based on the definition of Riesz–Riemann–Liouville fractional derivative, we study general fractional dynamical equations, construct its fractional Jacobi last multiplier, and, respectively, give the determining equation and three important properties of the multiplier. And then, we present the fractional Jacobi last multiplier method, which includes three theorems of finding conserved quantities of fractional dynamical systems. Further, in fractional framework, we explore the relation between Lie symmetry and Jacobi last multiplier. Furthermore, the fractional Jacobi last multiplier method is applied to the fractional generalized Hamiltonian system, the complete fractional Hamiltonian system, the standard fractional Hamiltonian system, the fractional Nambu system and the fractional Birkhoffian system, and five corresponding propositions are given. Also by using the fractional Jacobi last multiplier method, we respectively find the conserved quantities of a fractional relativistic Buchdahl model, a fractional Euler–Poinsot model and a fractional Duffing oscillator model. This work constructs a basic theoretical framework of fractional Jacobi last multiplier method, and provides a general method of fractional dynamics.

Noether’s theorem for fractional Birkhoffian systems of variable order

The Noether symmetries and conserved quantities of the Birkhoffian systems in terms of fractional derivatives of variable order are studied. Firstly, the Pfaff–Birkhoff–d’Alembert principle within fractional derivatives of variable order is obtained, and corresponding variable order fractional Birkhoff’s equations are deduced. Secondly, the invariance of the fractional Pfaff action of variable order is studied under the one-parameter group of infinitesimal transformations, and the definition of the variable order fractional conserved quantity is given. Finally, the Noether’s theorem for the fractional Birkhoffian system of variable order is established. At the end of this paper, an example is given to illustrate the application of the results.

Noether symmetries and conserved quantities for fractional Birkhoffian systems with time delay

The Noether symmetries and the conserved quantities for fractional Birkhoffian systems with time delay in terms of Riemann–Liouville fractional derivatives are proposed and studied. First, the fractional Pfaff–Birkhoff principle with time delay is proposed, and the fractional Birkhoff's equations with time delay are obtained. Second, based on the invariance of the fractional Pfaff action with time delay under a group of infinitesimal transformations, the Noether symmetric transformations and the Noether quasi-symmetric transformations of the system are defined, and the criteria of the Noether symmetries are established. Finally, the relationship between the symmetries and the conserved quantities are studied, and the Noether theorems for fractional Birkhoffian systems with time delay are established. Some examples are given to illustrate the application of the results.

Fractional Nambu dynamics

In this paper, we present a new fractional dynamical theory, i.e., the dynamics of a Nambu system with fractional derivatives (the fractional Nambu dynamics), and study its applications to mechanical systems. By using the definition of combined fractional derivative, we construct unified fractional Nambu equations and, respectively, give four new kinds of fractional Nambu equations based on the different definitions of fractional derivative. Further, we study the relations between the fractional Nambu system, the fractional generalized Hamiltonian system, the fractional Birkhoffian system, the fractional Hamiltonian system, the fractional Lagrangian system and a series of integer-order dynamical systems and present the transformation conditions. And furthermore, as applications of the fractional Nambu method, we construct two kinds of fractional dynamical models, which include the fractional Euler–Poinsot model of rigid body which rotates with respect to a fixed point and the fractional Yamaleev oscillator model. This work provides a general method for studying a fractional dynamical problem which is related to science and engineering.

Noether symmetries and conserved quantities for fractional Birkhoffian systems

This paper presents the variational problems for fractional Birkhoffian systems, and its corresponding symmetries and conserved quantities are further studied. First, the fractional Pfaff variational problems in terms of Riemann–Liouville fractional derivatives are proposed, and the fractional Pfaff–Birkhoff–d’Alembert principle is established. Thus, the fractional Birkhoff’s equations are derived. Second, we derive two basic formulae for the variation of Pfaff action and give the definitions and criteria of fractional Noether symmetries and quasi-symmetries. Third, we establish the Noether theorems of fractional Birkhoffian systems which reveal the relationship between fractional symmetries and fractional conserved quantities. Some special cases are discussed, showing the universality of our results. Two examples are analyzed to illustrate the applications of the results in detail.

Stability for manifolds of the equilibrium state of fractional Birkhoffian systems

In the paper, we present a new stability theory of fractional dynamics, i.e., the stability for manifolds of equilibrium state of a fractional Birkhoffian system, in terms of Riesz derivatives, and explore its applications. For an autonomous fractional Birkhoffian system, the gradient representation and second-order gradient representation are studied, and the conditions under which the system can be considered as a gradient system and a second-order gradient system are given, respectively. Then, equilibrium equations, perturbation equations and first approximate equations are obtained, and the stability theorem for manifolds of equilibrium state of the general autonomous system is used to a fractional Birkhoffian system, and three criterions on the stability for manifolds of the equilibrium state of the system are investigated. As special cases of the fractional Birkhoffian method, the three criterions can also be applied to a traditional Birkhoffian system, a fractional Hamiltonian system and a classical Hamiltonian system, respectively. As applications, by using the fractional Birkhoffian method of this paper, we construct three kinds of fractional dynamical models, which include a four-dimensional fractional dynamical model, a fractional Henon–Heiles model and a fractional Lorentz–Dirac model, and we explore the stability for manifolds of the equilibrium state of these models, respectively. This work provides a general method for studying the dynamic stability of an actual fractional dynamical model that is related to science and engineering.

Noether's theorems of a fractional Birkhoffian system within Riemann—Liouville derivatives

The Noether symmetry and the conserved quantity of a fractional Birkhoffian system are studied within the Riemann—Liouville fractional derivatives. Firstly, the fractional Birkhoff's equations and the corresponding transversality conditions are given. Secondly, from special to general forms, Noether's theorems of a standard Birkhoffian system are given, which provide an approach and theoretical basis for the further research on the Noether symmetry of the fractional Birkhoffian system. Thirdly, the invariances of the fractional Pfaffian action under a special one-parameter group of infinitesimal transformations without transforming the time and a general one-parameter group of infinitesimal transformations with transforming the time are studied, respectively, and the corresponding Noether's theorems are established. Finally, an example is given to illustrate the application of the results.