Discussion on Perturbation to Weak Noether Symmetry and Adiabatic Invariants for Lagrange Systems

We study a new symmetric perturbation, i.e. weakly Noether symmetric perturbation （WNSP）. The criterion and definition of WNSP and Noether symmetric perturbation （NSP） are given. A theorem between WNSP and adiabatic invariants is established. It is concluded that WNSP is different from NSP, the sufficient condition when WNSP is NSP can be presented, and the former is broader. We apply our results to the planar Kepler problem.     

Perturbation to Mei Symmetry and Generalized Mei Adiabatic Invariants for Nonholonomic Systems in Terms of Quasi-Coordinates

By introducing the coordination function f, the generalized Mei conserved quantities for the nonholonomic systems in terms of quasi-coordinates are given. Then based on the concept of adiabatic invariant, the perturbation to Mei symmetry and the generalized Mei adiabatic invariants for nonholonomic systems in terms of quasi-coordinates are studied.     

Generalized Mei Conserved Quantity of Mei Symmetry for Mechanico-electrical Systems with Nonholonomic Controllable Constraints

On the basis of the total time derivative along the trajectory, we study the generalized Mei conserved quantity of Mei symmetry for mechanico-electrieal systems with nonholonomic controllable constraints. Firstly, the definition and criterion of Mei symmetry for mechanico-electrical systems with nonholonomie controllable constraints are presented. Secondly, a coordination function is introduced, and the conditions of existence of generalized Mei conserved quantity as well as the forms are proposed. Lastly, an example is given to illustrate the application of the results.     

Lie Symmetries, Perturbation to Symmetries and Adiabatic Invariants of a Generalized Birkhoff System

We study the perturbation to symmetries and adiabatic invariants of a generalized Birkhoff system. Based on the invariance of differential equations under infinitesimal transformations, Lie symmetries, laws of conservations, perturbation to the symmetries and adiabatic invariants of the generalized Birkhoff system are presented. First, the concepts of Lie symmetries and higher order adiabatic invariants of the generalized Birkhoff system are proposed. Then, the conditions for the existence of the exact invariants and adiabatic invariants are proved, and their forms are given. Finally, an example is presented to illustrate the method and results.     

Special Lie Symmetry and Hojman Conserved Quantity of Appell Equations for a Holonomic System

Special Lie symmetry and Hojman conserved quantity of Appell equations for a holonomic system are studied. Appell equations and differential equations of motion for holonomic mechanic systems are established. Under special Lie infinitesimal transformations in which the time is invariable, the determining equation of the special Lie symmetry and the expressions of Hojman conserved quantity for Appell equations of holonomic systems are presented. Finally, an example is given to illustrate the application of the results.     

Lie Symmetrical Perturbation and Adiabatic Invariants of Generalized Hojman Type for Relativistic Birkhoffian Systems

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.     

A New Conserved Quantity Corresponding to Mei Symmetry of Tzenoff Equations for Nonholonomic Systems

A new conserved quantity is investigated by utilizing the definition and discriminant equation of Mei symmetry of Tzénoff equations for nonholonomic systems. In addition, the expression of this conserved quantity, and the determining condition induced new conserved quantity are also presented.     

A New Type of Non-Noether Adiabatic Invariants for Disturbed Lagrangian Systems： Adiabatic Invariants of Generalized Lutzky Type

For a Lagrangian system with the action of small disturbance, the Lie symmetrical perturbation and a new type of non-Noether adiabatic invariant are presented in general infinitesimal transformation groups. On the basis of the invariance of disturbed Lagrangian systems under general infinitesimal transformations, the determining equations of Lie symmetries of the system are constructed. Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of adiabatic invariant, i.e. generalized Lutzky adiabatic invariants, of a disturbed Lagrangian system are obtained by investigating the perturbation of Lie symmetries t＇or a Lagrangian system with the action of small disturbance. Finally, an example is given to illustrate the application of the method and results.     

Mei Symmetry and New Conserved Quantity of Tzenoff Equations for Holonomic Systems

A new conserved quantity is deduced from Mei symmetry of Tzenoff equations for holonomic systems. The expression of this new conserved quantity is given, and the determining equation to induce this new conserved quantity is presented. The results exhibit that this new method is easier to find more conserved quantities than the previously reported ones. Finally, application of this new result is presented by a practical example.     

Conservation Laws for Partially Conservative Variable Mass Systems via d＇Alembert＇s Principle

Conservation laws for partially conservative variable mass dynamical systems under symmetric infinitesimal transformations are determined. A generalization of Lagrange-d ＇Alembert＇s principle for a variable mass system in terms of asynchronous virtual variation is presented. The generalized Killing equations are obtained such that their solution yields the transformations and the associated conservation laws. An example illusstrative of the theory is furnished at the end as well.     

Perturbation to Lie Symmetry and Generalized Hojman Adiabatic Invariants for Relativistic Hamilton System

Based on the concept of higher-order adiabatic invariants of mechanical system with action of a small perturbation, the perturbation to Lie symmetry and generalized Hojman adiabatic invariants for the relativistic Hamilton system are studied. Perturbation to Lie symmetry is discussed under general infinitesimal transformation of groups in which time is variable. The form and the criterion of generalized Hojman adiabatic jnvariants for this system are obtained. Finally, an example is given to illustrate the results.     

Structural Equation and Mei Conserved Quantity of Mei Symmetry for Appell Equations with Redundant Coordinates

Structural equations and Mei conserved quantity of Mei symmetry for Appell equations in a holonomic system with redundant coordinates are studied. Some aspects, including the differential equations of motion, the definition and the criterion of Mei symmetry, the form of structural equations and Mei conserved quantity of Mei symmetry of Appell equations for a holonomic system with redundant coordinates, are also investigated. Finally, an example is given to illustrate the application of the results.     

Conformal Invariance and Conserved Quantities of General Holonomic Systems

Conformed invariance and conserved quantities of general holonomic systems are studied. A one-parameter infinitesimal transformation group and its infinitesimal transformation vector of generators are described. The definition of conformal invariance and determining equation for the system are provided. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The necessary and suttlcient condition, that conformal invariance of the system would be Lie symmetry, is obtained under the infinitesimal one-parameter transformation group. The corresponding conserved quantity is derived with the aid of a structure equation. Lastly, an example is given to demonstrate the application of the result.     

Perturbation to Lie Symmetry and Hojman Exact and Adiabatic Invariants of Generalized Raitzin Canonical Equation of Motion

In this paper, firstly, we get the Hojman exact invariants by Lie symmetry for an undisturbed generalized Raitzin equation of motion. Secondly, we study the perturbation to Lie symmetry of generalized Raitzin canonical equation of motion and get Hojman adiabatic invariants. Lastly, an example is given to illustrate the application of the results.     

Perturbation and Adiabatic Invariants of Mei Symmetry for Nonholonomic Mechanical Systems

Based on the concept of adiabatic invariant, the perturbation and adiabatic invariants of the Mei symmetry for nonholonomic mechanical systems are studied. The exact invariants of the Mei symmetry for the system without perturbation are given. The perturbation to the Mei symmetry is discussed and the adiabatic invariants of the Mei symmetry for the perturbed system are obtained.     

Perturbation to Lie Symmetry and Lutzky Adiabatic Invariants for Lagrange Systems

Based on the concept of adiabatic invariant, perturbation to Lie symmetry and Lutzky adiabatic invariants for Lagrange systems are studied by using different methods from those of previous works. Exact invariants induced from Lie symmetry of the system without perturbation are given. Perturbation to Lie symmetry is discussed and Lutzky adiabatic invariants of the system subject to perturbation are obtained.     

Perturbation of Symmetries and Hojman Adiabatic Invariants for Mechanical Systems with Unilateral Holonomic Constraints

The perturbation of symmetries and adiabatic invariants for mechanical systems with unilateral holonomic constraints are studied. The exact invariant in the form of Hojman led by special Lie symmetries for an undisturbed system with unilateral constraints is given. Based on the concept of high-order adiabatic invariant of mechanical systems, the perturbation of Lie symmetries for the system under the action of small disturbance is investigated, and a new adiabatic invariant for the system with unilateral holonomic constraints is obtained, which can be called Hojman adiabatic invariant. In the end of the paper, an example is given to illustrate the application of the results.     

Perturbation of Symmetries and Mei Adiabatic Invariants of Nonholonomic Systems with Servoconstraints

The perturbation of symmetries and Mei adiabatic invariants of nonholonomic systems with servoconstraints are studied. The exact invariants in the form of Mei conserved quantities introduced by the Mei symmetry of nonholonomic systems with servoconstraints without perturbations are given. Based on the definition of higher-order adiabatic invariants of mechanical systems, the perturbation of Mei symmetries for nonholonomic .systems with servoconstraints under the action of small disturbance is investigated, and Mei adiabatic invatiants of the system are obtained. An example is given to illustrate the application of the results.     

Perturbation to Lie Symmetry and Adiabatic Invariants for General Holonomic Mechanical Systems

Based on the concept of adiabatic invariant, the perturbation to the Lie symmetry and adiabatic invariants for general holonomic mechanical systems are studied. The exact invariants induced directly from the Lie symmetry of the system without perturbation are given. The perturbation to the Lie symmetry is discussed and the adiabatic invariants that have the different form from that in [Act. Phys. Sin. 55 (2006) 3236 (in Chinese)] of the perturbed system, are obtained.