Abstract:
Based on the theory of symmetries and conserved quantities of the singular Lagrange system, the perturbations to the symmetries and adiabatic invariants of the singular Lagrange systems are discussed. Firstly, the concept of higher-order adiabatic invariants of the singular Lagrange system is 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 these results.

Abstract:
the main objective of this thesis is the development of parameter representations which are interconnected in flexible rotor and autobalance systems coupled. this goal consists in achieving independent set of differential equations for each section of the shaft and applying stiffness method to obtain a global matrix. these models have been used to analyze elastic and inertial properties of rotors. after we obtain a global matrix, we can establish a set of differential equations, which may be simulated by software and analyzed stability and dynamical behavior of our complete system. inertial associated coefficients have been founded in this model, by the way; effects by each section and connections have been gotten. dynamical behavior of balls and rotor has been analyzed by lagrange method to get a differential equations system. in the case of a single rotor and a single drum (with balls) in different plane, an experimental model was developed; theoretical (software) and experimental results were similar, and this showed the efficiency of the balancing method. there are several conditions of mass, stiffness and frequency which system works, these characteristics were founded.

Abstract:
This paper is concerned with the numerical solution for singular perturbation system of two coupled second ordinary differential equations with initial and boundary conditions, respectively. Fitted finite difference scheme on a uniform mesh, whose solution converges pointwise independently of the singular perturbation parameter is constructed and analyzed.

This work deals with the
numerical solution of singular perturbation system of ordinary differential
equations with boundary layer. For the numerical solution of this problem
fitted finite difference scheme on a uniform mesh is constructed and analyzed.
The uniform error estimates for the approximate solution are obtained.

A glance at Bessel
functions shows they behave similar to the damped sinusoidal function. In this
paper two physical examples (pendulum and spring-mass system with linearly
increasing length and mass respectively) have been used as evidence for this
observation. It is shown in this paper how Bessel functions can be approximated
by the damped sinusoidal function. The numerical method that is introduced
works very well in adiabatic condition (slow change) or in small time
(independent variable) intervals. The results are also compared with the
Lagrange polynomial.

Abstract:
A kind of conserved quantity which is directly induced by Mei symmetry of Lagrange system is studied. The definition and criterion of Mei symmetry for Lagrange system are given. The condition under which Mei symmetry can lead to a conserved quantity and the form of the conserved quantity are obtained. An example is given to illustrate the application of the result.

Abstract:
The Noether symmetry and Lie symmetry of the Lagrange system subjected to gyroscopic forces are studied. The condition that the system, under gyroscopic forces , can keep its Noether symmetry and Noether conserved quantity is given. And the condition that the system subjected to gyroscopic forces can keep its Lie symme try and Hojman conserved quantity is also given. Finally, two examples are given to illustrate the application of the results.

Abstract:
In this paper,the polynomial type Lagrange equation of finite dimensional constrained dynamics is considered,and four possibilities of Euler-Lagrange equations possibilities are analysed.Based on Wu elimination method and Wu differential characteristic set method,two algorithms are presented for determining the four possibilities.Using the two algorithms,without calculating the rank of Hessian matrix,which case of the four possibilities of the Euler-Lagrange equations is determined,and the corresponding results are obtained.On the symbolic computation software platform,the two algorithms can be executed in computers.

Abstract:
In this paper, the Lie-form invariance of the Lagrange system is studied. The definition and the criterion of the Lie-form invariance of the Lagrange system are given. The Hojman conserved quantity and a new type of conserved quantity deduced from the Lie-form invariance are obtained. Finally, two examples are presented to illustrate the application of the results.