Perturbation to Symmetries and Adiabatic Invariants of Nonholonomic System in Terms of Quasi-coordinates

Based on the theory of Lie symmetries and conserved quantities, the exact invariants and adiabatic invariants of nonholonomic system in terms of quasi-coordinates are studied. The perturbation to symmetries for the nonholonomic system in terms of quasi-coordinates under small excitation is discussed. The concept of high-order adiabatic invariant is presented, and the forms of exact invariants and adiabatic invariants as well as the conditions for their existence are given. Then the corresponding inverse problem is studied.     

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

Abstract Based on the concept of adiabatic invariant, the perturbation to Mei symmetry and adiabatic invariants for nonholonomic mechanical systems in terms of quasi-coordinates are studied. The definition of the perturbation to Mei symmetry for the system is presented, and the criterion of the perturbation to Mei symmetry is given. Meanwhile, the Mei adiabatic invariants for the perturbed system are obtained.     

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

Based on the concept of adiabatic invariant, the perturbation to Meisymmetry and adiabatic invariants for nonholonomic mechanical systems interms of quasi-coordinates are studied. The definition of the perturbationto Mei symmetry for the system is presented, and the criterion of theperturbation to Mei symmetry is given. Meanwhile, the Mei adiabaticinvariants for the perturbed system are obtained.     

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.     

Exact Invariants and Adiabatic Invariants of Raitzin's Canonical Equations of Motion for Nonholonomic System of Non-Chetaev's Type

The exact invariants and the adiabatic invariants of Raitzin's canonical equations of motion for the nonholonomic system of non-Chetaev's type 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 mechanical system with the action of a small perturbation, the form 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.     

Perturbation to Noether-Mei Symmetry and Adiabatic Invariants for Nonholonomic Mechanical Systems in Phase Space

Based on the concept of adiabatic invariant, the perturbation to Noether Mei symmetry and adiabatic invariants for nonholonomie mechanical systems in phase space are studied. The definition of the perturbation to Noether-Mei symmetry for the system is presented, and the criterion of the perturbation to Noether-Mei symmetry is given. Meanwhile, the Noether adiabatic invariants and the Mei adiabatic invariants for the perturbed system are obtained.     

Mei Adiabatic Invariants Induced by Perturbation of Mei Symmetry for Nonholonomic Controllable Mechanical Systems

Two types of Mei adiabatic invariants induced by perturbation of Mei symmetry for nonholonomic controllable mechanical systems are reported. Criterion and restriction equations determining Mei symmetry after being disturbed of the system are established. Form and existence condition of Mei adiabatic invariants 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.     

准坐标下完整系统的精确不变量与绝热不变量

基于对称性与不变量理论，研究了准坐标下完整系统的精确不变量与绝热不变量。研究了在小扰动作用下准坐标完整系统的对称性摄动。提出了高阶绝热不变量的概念，给出了精确不变量与绝热不变量存在的形式与条件。然后研究了相应的逆问题。     

Hojman Exact Invariants and Adiabatic Invariants of Hamilton System

The perturbation to Lie symmetry and adiabatic invariants are studied. Based on the concept of higher-order adiabatic invariants of mechanical systems with action of a small perturbation, the perturbation to Lie symmetry is studied, and Hojman adiabatic invariants of Hamilton system are obtained. An example is given to illustrate the application of the results.     

Mei Adiabatic Invariants Induced by Perturbation of Mei Symmetry forNonholonomic Controllable Mechanical Systems

Two types of Mei adiabatic invariants induced by perturbation of Mei symmetry for nonholonomic controllable mechanical systems are reported. Criterion and restriction equations determining Mei symmetry after being disturbed of the system are established. Form and existence condition of Mei adiabatic invariants are obtained.     

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.     

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 to Lie-Mei Symmetry and Adiabatic Invariants for Birkhoffian Systems

Based on the concept of adiabatic invariant, the perturbation to Lie-Mei symmetry and adiabatic invariants for Birkhoffian systems are studied. The definition of the perturbation to Lie-Mei symmetry for the system is presented, and the criterion of the perturbation to Lie-Mei symmetry is given. Meanwhile, the Hojman adiabatic invariants and the Mei adiabatic invariants for the perturbed system are obtained.     

Perturbation to Lie-Mei Symmetry and Adiabatic Invariants for Birkhoffian Systems

Based on the concept of adiabatic invariant, the perturbation to Lie-Mei symmetry and adiabatic invariants for Birkhoffian systems are studied. The definition of the perturbation to Lie-Mei symmetry for the system is presented, and the criterion of the perturbation to Lie-Mei symmetry is given. Meanwhile, the Hojman adiabatic invariants and the Mei adiabatic invariants for the perturbed system are obtained.     

Perturbation to Unified Symmetry and Adiabatic Invariants for Relativistic Hamilton Systems

Based on the concept of adiabatic invariant, the perturbation to unified symmetry and adiabatic invariants for relativistic Hamilton systems are studied. The definition of the perturbation to unified symmetry for the system is presented, and the criterion of the perturbation to unified symmetry is given. Meanwhile, the Noether adiabatic invariants, the generalized Hojman adiabatic invariants, and the Mei adiabatic invariants for the perturbed system are obtained.     

Perturbation to Mei Symmetry and Noether Adiabatic Invariants for Birkhoffian Systems

Based on the concept of adiabatic invariant, the perturbation to Mei symmetry and Noether adiabatic invariants for Birkhoffian systems are studied. The exact invariants of Mei symmetry for the system without perturbation are given. The perturbation to Mei symmetry is discussed and the Noether adiabatic invariants induced from the perturbation to Mei symmetry of the system are obtained.     

Perturbation to symmetries and adiabatic invariants of a type of nonholonomic singular system

Based on the theory of symmetries and conserved quantities, the perturbation to the symmetries and adiabatic invariants of a type of nonholonomic singular system are discussed. Firstly, the concept of higher order adiabatic invariants of the system is proposed. Secondly, the conditions for existence of the exact invariants and adiabatic invariants are proved, and their forms are given. Thirdly, we study the inverse problems of the perturbation to symmetries of the system. An example is presented to illustrate these 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.