Wu Hui-Bin

Form invariance and new conserved quantity of generalised Birkhoffian system

A form invariance and a conserved quantity of the generalised Birkhoffian system are studied. First, a definition and a criterion of the form invariance are given. Secondly, through the form invariance, a new conserved quantity can be deduced. Finally, an example is given to illustrate the application of the result.

Lagrange–Noether method for solving second-order differential equations

The purpose of this paper is to provide a new method called the Lagrange–Noether method for solving second-order differential equations. The method is, firstly, to write the second-order differential equations completely or partially in the form of Lagrange equations, and secondly, to obtain the integrals of the equations by using the Noether theory of the Lagrange system. An example is given to illustrate the application of the result.

Stability with respect to partial variables for Birkhoff systems

In this paper, the stability with respect to partial variables for the Birkhoff system is studied. By transplanting the results of the partial stability for general systems to the Birkhoff system and constructing a suitable Liapunov function, the partial stability of the system can be achieved. Finally, two examples are given to illustrate the application of the results.

Bifurcation for the generalized Birkhoffian system

The system described by the generalized Birkhoff equations is called a generalized Birkhoffian system. In this paper, the condition under which the generalized Birkhoffian system can be a gradient system is given. The stability of equilibrium of the generalized Birkhoffian system is discussed by using the properties of the gradient system. When there is a parameter in the equations, its influences on the stability and the bifurcation problem of the system are considered.

Generalized Hamilton system and gradient system

The generalized Hamilton system and the gradient system are two different important dynamical systems. The relation between the two systems is studied. The generalized Hamilton system is a generalization of the Hamilton system. A Birkhoff system can become a generalized Hamilton system under some conditions. The gradient system and its application are given. The relation of the two systems is obtained. Some examples are given to illustrate the application of the result.

Symmetry of Lagrangians of holonomic systems in terms of quasi-coordinates

This paper discusses the symmetry of Lagrangians of holonomic systems in terms of quasi-coordinates. Firstly, the definition and the criterion of the symmetry are given. Secondly, the condition under which there exists a conserved quantity and the form of the conserved quantity are obtained. Finally, an example is shown to illustrate the application of the results.

Skew-gradient representation of generalized Birkhoffian system*

The skew-gradient representation of a generalized Birkhoffian system is studied. A condition under which the generalized Birkhoffian system can be considered as a skew-gradient system is obtained. The properties of the skew-gradient system are used to study the properties, especially the stability, of the generalized Birkhoffian system. Some examples are given to illustrate the application of the result.

Hamilton-Jacobi method for solving ordinary differential equations *

The Hamilton?Jacobi method for solving ordinary differential equations is presented in this paper. A system of ordinary differential equations of first order or second order can be expressed as a Hamilton system under certain conditions. Then the Hamilton?Jacobi method is used in the integration of the Hamilton system and the solution of the original ordinary differential equations can be found. Finally, an example is given to illustrate the application of the result.

Lie-form invariance of the Lagrange system

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.

Symmetry of Lagrangians of nonholonomic systems of non-Chetaev’s type

This paper studies the symmetry of Lagrangians of nonholonomic systems of non-Chetaev’s type. First, the denition and the criterion of the symmetry of the system are given. Secondly, it obtains the condition under which there exists a conserved quantity and the form of the conserved quantity. Finally, an example is shown to illustrate the application of the result.