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In this paper, we present a new method of fractional dynamics, i.e., the fractional Mei symmetrical perturbation method of a disturbed system, and explore the adiabatic invariant directly led by the perturbation. For a dynamical system which is disturbed by small forces of perturbation, the disturbed fractional generalized Hamiltonian equation is investigated and, under a more general kind of fractional infinitesimal transformation of a Lie group, the fractional Mei symmetrical definition and determining equation of a disturbed dynamical system are given; then, the determining equation of fractional Mei symmetrical perturbation is obtained. In particular, we present the fractional Mei symmetrical perturbation method of a disturbed dynamical system and it is found that, using the new method, we can find a new kind of non-Noether adiabatic invariant directly led by the perturbation. As special cases, we obtain a new fractional Mei symmetrical conservation law of the undisturbed dynamical system and a Mei symmetrical perturbation theorem of the disturbed integer dynamical system. Also, as the fractional Mei symmetrical perturbation method’s applications, we find the adiabatic invariants of a disturbed fractional Duffing oscillator and a disturbed fractional Lotka biochemical oscillator. This work constructs a basic theoretical framework of fractional Mei symmetrical perturbation method and provides a general method of fractional dynamics. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we present a new type of fractional Lie symmetrical method for finding conserved quantities and explore its applications. For the fractional generalized Hamiltonian system, we introduce a new kind of single-parameter fractional infinitesimal transformation of Lie group in α−1 order space and, under this transformation, give the invariance of the fractional dynamical system and the fractional Lie symmetrical determining equation. Further, a number of important relationships of the fractional Lie symmetrical method are investigated, which reveal the interior properties of the system. By using these relationships, a fractional Lie symmetrical basic integral variable relation and a new fractional Lie symmetrical conservation law are presented. The new conserved quantity is constructed base on fractional Lie symmetrical infinitesimal generators and the interior properties of the system itself, without solving the complicated structural equation. Furthermore, the fractional Lie symmetrical method is applied to the fractional generalized Hamiltonian system of even dimensions. Also, as the new fractional Lie symmetrical method's applications, we respectively find the conserved quantities of a fractional Duffing oscillator model and a fractional Lotka biochemical oscillator model. [ABSTRACT FROM AUTHOR]

In this paper, we present the fractional Mei symmetrical method of finding conserved quantity and explore its applications to physics. For the fractional generalized Hamiltonian system, we introduce the fractional infinitesimal transformation of Lie groups and, under the transformation, give the fractional Mei symmetrical definition, criterion and determining equation. Then, we present the fractional Mei symmetrical theorem of finding conserved quantity. As the fractional Mei symmetrical method's applications, we respectively find the conserved quantities of a fractional general relativistic Buchduhl model, a fractional three-body model and a fractional Robbins-Lorenz model. [ABSTRACT FROM AUTHOR]

For a dynamical system in phase space, a new Lie symmetrical method to find a conserved quantity is presented in a general infinitesimal transformation of Lie groups. Based on the invariance of the differential equations of motion for the system under a general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations are obtained. Then, an important relationship that reveals the interior properties of a dynamical system in phase space is given. By using the relationship, a Lie symmetrical basic integral variable relation and a new Lie symmetrical conservation law for the dynamical system in phase space are given. The new conserved quantity is constructed in terms of the infinitesimal generators of Lie symmetry and the system itself without solving the structural equation. Furthermore, the method is applied in the Hamiltonian system, the nonconservative Hamiltonian system and the nonholonomic Hamiltonian system. Finally, one example is given to illustrate the method and results of the application. [ABSTRACT FROM AUTHOR]

A form invariance of the relativistic Birkhoffian system is studied, and the conserved quantities of the system are obtained. Under the infinitesimal transformation of groups, the definition and criteria of the form invariance of the system were given. In view of the invariance of relativistic Pfaff-Birkhoff-D' Alembert principle under the infinitesimal transformation of groups, the theory of Noethe, symmetries of the relativistic Birkhoffian system were constructed. The relation between the form invariance and the Noether symmetry is studied, and the results show that the form invariance can also lead to the Noether symmetrical conserved quantity of the relativistic Birkhoffian system under certain conditions. [ABSTRACT FROM AUTHOR]

The form invariance of constrained Birkhoffian system is a kind of invariance of the constrained Birkhoffian equations under infinitesimal transformations. The definition and criteria of the form invariance of constrained Birkhoffian system are given, and the relation of the form invariance and the Noether symmetry is studied. [ABSTRACT FROM AUTHOR]