Binding System Of A New Type Of Adiabatic Invariants

The exact invariants and adiabatic invariants for constrained mechanical systems, which can be found through symmetries of the system, play an important role in the fields of mathematics, physics and mechanics. Perturbation method is widely used in celestial mechanics, physics and mechanics as well. As we know, even a small perturbation may influence the original symmetries and exact invariants of the constrained mechanical system. Under the action of small perturbation, the exact invariants turn into adiabatic invariants. In addition, the exact invariants and adiabatic invariants of Hojman type are dependent on not only infinitesimal generators of Lie symmetry, but also structural equations which are very difficult to solve. Therefore, it is important to find new exact invariants and adiabatic invariants which are dependent on infinitesimal generators of Lie symmetry and the system itself only. Thus, this dissertation treats a new type of exact invariants and adiabatic invariants for constrained mechanical systems.In the first chapter, we survey briefly the latest progress in the theory of constrained mechanical systems and symmetrical perturbation.In the second chapter, the author proposes a new type of exact invariants and adiabatic invariants for constrained mechanical systems in phase space. Firstly, by introducing the general Lie group of transformations that the variation of time is considered, the author derives the determining equations of Lie symmetry and a new type of exact invariants for the system in phase space. Secondly, the result is successfully used in Hamilton system. Thirdly, according to the theory of perturbation, the author gets a new type of adiabatic invariants for the system in phase space and obtains a new type of adiabatic invariants for Hamilton system correspondingly by applying the result in Hamilton system. Finally, one example is given to illustrate the method and results.In chapter three, we study a new type of exact invariants and adiabatic invariants for generalized Hamilton system. Firstly, by introducing the general Lie group of transformations, the author derives the determining equations of Lie symmetry and a new type of exact invariants for the generalized Hamilton system. Secondly, an important relation of even dimension generalized Hamilton system is investigated. Thirdly, according to the theory of perturbation, the author gets a new type of adiabatic invariants for the generalized Hamilton system. Finally, an example is presented to illustrate the method and results.In chapter four, we study a new type of exact invariants and adiabatic invariants for Birkhoff system and generalized Birkhoff system. Firstly, by introducing the general Lie group of transformations, the author obtains the determining equations of Lie symmetry and a new type of exact invariants for the Birkhoff system. Secondly, an important relation of Birkhoff system is investigated. Thirdly, according to the theory of perturbation, the author gets a new type of adiabatic invariants for the Birkhoff system. Furthermore, the author applies the method into generalized Birkhoff system and obtains a new type of exact invariants and adiabatic invariants. Finally, an example is given to illustrate the method and results.In chapter five, the author presents the action of small forces of perturbation, Lie symmetries, symmetrical perturbation and a new type of exact invariants and 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, several deductions are given in the special infinitesimal transformations. Finally, one example is given to illustrate the method and results of the application.In chapter six, the author concludes the dissertation by summarizing the results and presents several ideas in the further work.