Infinitesimal Transformation Of Lie Groups Research Articles

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.

Symmetry and conserved quantities for non-material volumes

This paper investigates the Lie symmetry and conserved quantities of non-material volumes. The Lie symmetrical determining equations of the system are presented by introducing the invariance of equations of motion for the system under general infinitesimal transformation of Lie groups. The structure equations and the form of conserved quantities are calculated. And three kinds of conserved quantities, i.e., Noether, Lutzky and Mei conserved quantities of the systems, are derived. In addition, the Hojman conserved quantity of the systems is proposed under the special infinitesimal transformations. An example is given to illustrate the application of the method and result, and four kinds of conserved quantities are obtained under the Lie symmetrical transformations.

Basic Theory of Fractional Conformal Invariance of Mei Symmetry and its Applications to Physics

In this paper, we present a basic theory of fractional dynamics, i.e., the fractional conformal invariance of Mei symmetry, and find a new kind of conserved quantity led by fractional conformal invariance. For a dynamical system that can be transformed into fractional generalized Hamiltonian representation, we introduce a more general kind of single-parameter fractional infinitesimal transformation of Lie group, the definition and determining equation of fractional conformal invariance are given. And then, we reveal the fractional conformal invariance of Mei symmetry, and the necessary and sufficient condition whether the fractional conformal invariance would be the fractional Mei symmetry is found. In particular, we present the basic theory of fractional conformal invariance of Mei symmetry and it is found that, using the new approach, we can find a new kind of conserved quantity; as a special case, we find that an autonomous fractional generalized Hamiltonian system possesses more conserved quantities. Also, as the new method’s applications, we, respectively, find the conserved quantities of a fractional general relativistic Buchduhl model and a fractional Duffing oscillator led by fractional conformal invariance of Mei symmetry.

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.

A New Type of Fractional Lie Symmetrical Method and its Applications

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.

A New Method of Fractional Dynamics, i.e., Fractional Mei Symmetrical Method for Finding Conserved Quantity, and its Applications to Physics

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.

A Lie symmetrical basic integral variable relation and a new conservation law for generalized Hamiltonian systems

For a generalized Hamiltonian system, 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. And a number of important relationships of the Lie symmetrical method for a generalized Hamiltonian system are investigated, which reveal the interior properties of the system. By using the relationships, a Lie symmetrical basic integral variable relation and a new Lie symmetrical conservation law for generalized Hamiltonian systems is presented. The basic integral variable relation not only can be used for a linear dynamic system but can also be used for a nonlinear dynamic system. The new conserved quantity is constructed in terms of infinitesimal generators of Lie symmetry and the interior properties of the system itself without solving the structural equation that may be very difficult to solve. Then, the method is applied in the generalized Hamiltonian system of even dimensions and the Hamiltonian system, respectively. Furthermore, the relationship between the generalized Hamiltonian system and a Birkhoffian system is studied, and a Lie symmetrical conservation law for a semiautonomous (or autonomous) Birkhoffian system is obtained. Finally, one example is given to illustrate the method and results of the application.

A new Lie symmetrical method of finding a conserved quantity for a dynamical system in phase space

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.

A new Lie symmetrical method of finding conserved quantity for Birkhoffian systems

For a Birkhoffian system, a new Lie symmetrical method to find a conserved quantity is given. Based on the invariance of the equations of motion for the system under a general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations are obtained. Then, several important relationships which reveal the interior properties of the Birkhoffian system are given. By using these relationships, a new Lie symmetrical conservation law for the Birkhoffian system is presented. The new conserved quantity is constructed in terms of infinitesimal generators of the Lie symmetry and the system itself without solving the structural equation which may be very difficult to solve. Furthermore, several deductions are given in the special infinitesimal transformations and the results are reduced to a Hamiltonian system. Finally, one example is given to illustrate the method and results of the application.

A new type of non-Noether exact invariants and adiabatic invariants of generalized Hamiltonian systems

For a generalized Hamiltonian system with the action of small forces of perturbation, the Lie symmetries, symmetrical perturbation, and adiabatic invariants is presented. Based on the invariance of equations of motion for the system under general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations, and exact invariants of the system are given. Then the determining equations of Lie symmetrical perturbation and adiabatic invariants of the disturbed systems are obtained. Furthermore, in the special infinitesimal transformations, two deductions are given. At the end of the paper, one example is given to illustrate the application of the method and result.

Lie symmetries, symmetrical perturbation and a new adiabatic invariant for disturbed nonholonomic systems

For a nonlinear nonholonomic constrained mechanical system with the action of small forces of perturbation, Lie symmetries, symmetrical perturbation, and a new type of non-Noether adiabatic invariants are presented in general infinitesimal transformation of Lie groups. Based on the invariance of the equations of motion for the system under general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations, constraints restriction equations, additional restriction equations, and exact invariants of the system are given. Then, under the action of small forces of perturbation, the determining equations, constraints restriction equations, and additional restriction equations of the Lie symmetrical perturbation are obtained, and adiabatic invariants of the Lie symmetrical perturbation, the weakly Lie symmetrical perturbation, and the strongly Lie symmetrical perturbation for the disturbed nonholonomic system are obtained, respectively. Furthermore, a set of non-Noether exact invariants and adiabatic invariants are given in the special infinitesimal transformations. Finally, one example is given to illustrate the application of the method and results.