Based on the invariance of differential equations under infinitesimal transformations, Lie symmetry, laws of conservations, perturbation to the symmetries and adiabatic invariants of Poincaré equations are presen...Based on the invariance of differential equations under infinitesimal transformations, Lie symmetry, laws of conservations, perturbation to the symmetries and adiabatic invariants of Poincaré equations are presented. The concepts of Lie symmetry and higher order adiabatic invariants of Poincaré equations are proposed. The conditions for existence of the exact invariants and adiabatic invariants are proved, and their forms are also given. In addition, an example is presented to illustrate these results.展开更多
The perturbations to symmetries and adiabatic invariants for nonconservative systems of generalized classical mechanics axe studied. The exact inwriant in the form of Hojman from a particular Lie symmetry for an undis...The perturbations to symmetries and adiabatic invariants for nonconservative systems of generalized classical mechanics axe studied. The exact inwriant in the form of Hojman from a particular Lie symmetry for an undisturbed system of generalized mechanics is given. Based on the concept of high-order adiabatic invaxiant in generalized mechanics, the perturbation to Lie symmetry for the system under the action of small disturbance is investigated, and a new adiabatic invaxiant for the nonconservative system of generalized classical mechanics is obtained, which can be called the Hojman adiabatic invaxiant. An example is also given to illustrate the application of the results.展开更多
The exact invariants and the adiabatic invariants of Raitzin's canonical equations of motion for a nonlinear nonholonomic mechanical system are studied. The relations between the invariants and the symmetries of the ...The exact invariants and the adiabatic invariants of Raitzin's canonical equations of motion for a nonlinear nonholonomic mechanical system are studied. The relations between the invariants and the symmetries of the system are established. Based on the concept of higher-order adiabatic invariant of a mechanical system under the action of a small perturbation, the forms of the exact invariants and adiabatic invariants and the conditions for their existence are proved. Finally, the inverse problem of the perturbation to symmetries of the system is studied and an example is also given to illustrate the application of the results.展开更多
For a relativistic Birkhoffian system, the Lie symmetrical perturbation and adiabatic invariants of generalized Bojman type are studied under general infinitesimal transformations. On the basis of the invariance of re...For a relativistic Birkhoffian system, the Lie symmetrical perturbation and adiabatic invariants of generalized Bojman type are studied under general infinitesimal transformations. On the basis of the invariance of relativistic Birkhotfian equations under general infinitesimal transformations,Lie symmetrical transformations of the system are constructed, which only depend on the Birkhoffian variables. The exact invariants in the form of generalized Hojman conserved quantities led by the Lie symmetries of relativistic Birkhoffian system without perturbations are given. Based on the definition of higher-order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for relativistic Birkhoffian system with the action of small disturbance is investigated, and a new type of adiabatic invariants of the system is obtained. In the end of the paper, an example is given to illustrate the application of the results.展开更多
Perturbation to Noether quasi-symmetry and adiabatic invariants for the nonholonomic system on time scales are studied. Firstly, some properties of time scale calculus are reviewed. Secondly, the differential equation...Perturbation to Noether quasi-symmetry and adiabatic invariants for the nonholonomic system on time scales are studied. Firstly, some properties of time scale calculus are reviewed. Secondly, the differential equations of motion for the nonholonomic system on time scales, Noether quasi-symmetry and conserved quantity are given. Thirdly, perturbation to Noether quasi-symmetry and adiabatic invariants, which are the main results of this paper, are investigated. The main results are achieved by two steps, the first step is to obtain adiabatic invariants without transforming the time, and the next is to obtain adiabatic invariants under the infinitesimal transformations of both the time and the coordinates. And in the end, an example is given to illustrate the methods and results.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant No 10372053) and the Natural Science Foundation of Henan Province, China (Grant No 0311010900).
文摘Based on the invariance of differential equations under infinitesimal transformations, Lie symmetry, laws of conservations, perturbation to the symmetries and adiabatic invariants of Poincaré equations are presented. The concepts of Lie symmetry and higher order adiabatic invariants of Poincaré equations are proposed. The conditions for existence of the exact invariants and adiabatic invariants are proved, and their forms are also given. In addition, an example is presented to illustrate these results.
基金Project supported by the Natural Science Foundation of High Education of Jiangsu Province, China (Grant No 04KJA130135).
文摘The perturbations to symmetries and adiabatic invariants for nonconservative systems of generalized classical mechanics axe studied. The exact inwriant in the form of Hojman from a particular Lie symmetry for an undisturbed system of generalized mechanics is given. Based on the concept of high-order adiabatic invaxiant in generalized mechanics, the perturbation to Lie symmetry for the system under the action of small disturbance is investigated, and a new adiabatic invaxiant for the nonconservative system of generalized classical mechanics is obtained, which can be called the Hojman adiabatic invaxiant. An example is also given to illustrate the application of the results.
基金Project supported by the Heilongjiang Natural Science Foundation of China (Grant No 9507).
文摘The exact invariants and the adiabatic invariants of Raitzin's canonical equations of motion for a nonlinear nonholonomic mechanical system are studied. The relations between the invariants and the symmetries of the system are established. Based on the concept of higher-order adiabatic invariant of a mechanical system under the action of a small perturbation, the forms of the exact invariants and adiabatic invariants and the conditions for their existence are proved. Finally, the inverse problem of the perturbation to symmetries of the system is studied and an example is also given to illustrate the application of the results.
基金The project supported by National Natural Science Foundation of China under Grant Nos. 10372053 and 10472040, the Natural Science Foundation of Hunan Province under Grant No. 03JJY3005, the Scientific Research Foundation of Eduction Department of Hunan Province under Grant No. 02C033 and the 0utstanding Young Talents Training Fund of Liaoning Province under Grant No. 309005
文摘For a relativistic Birkhoffian system, the Lie symmetrical perturbation and adiabatic invariants of generalized Bojman type are studied under general infinitesimal transformations. On the basis of the invariance of relativistic Birkhotfian equations under general infinitesimal transformations,Lie symmetrical transformations of the system are constructed, which only depend on the Birkhoffian variables. The exact invariants in the form of generalized Hojman conserved quantities led by the Lie symmetries of relativistic Birkhoffian system without perturbations are given. Based on the definition of higher-order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for relativistic Birkhoffian system with the action of small disturbance is investigated, and a new type of adiabatic invariants of the system is obtained. In the end of the paper, an example is given to illustrate the application of the results.
基金Supported by the National Natural Science Foundation of China(11802193,11572212)the Natural Science Foundation of the Jiangsu Higher Education Institutions of China(18KJB130005)+2 种基金the Jiangsu Government Scholarship for Overseas Studiesthe Science Research Foundation of Suzhou University of Science and Technology(331812137)the Natural Science Foundation of Suzhou University of Science and Technology
文摘Perturbation to Noether quasi-symmetry and adiabatic invariants for the nonholonomic system on time scales are studied. Firstly, some properties of time scale calculus are reviewed. Secondly, the differential equations of motion for the nonholonomic system on time scales, Noether quasi-symmetry and conserved quantity are given. Thirdly, perturbation to Noether quasi-symmetry and adiabatic invariants, which are the main results of this paper, are investigated. The main results are achieved by two steps, the first step is to obtain adiabatic invariants without transforming the time, and the next is to obtain adiabatic invariants under the infinitesimal transformations of both the time and the coordinates. And in the end, an example is given to illustrate the methods and results.