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 引用本文: 陈向炜, 张晔, 梅凤翔. 用具有负定非对称矩阵的梯度系统构造稳定的广义Birkhoff系统[J]. 力学学报, 2017, 49(1): 149-153.
Chen Xiangwei, Zhang Yey, Mei Fengxiang. STABLE GENERALIZED BIRKHOFF SYSTEMS CONSTRUCTED BY USING A GRADIENT SYSTEM WITH NON-SYMMETRICAL NEGATIVE-DEFINITE MATRIX[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(1): 149-153.
 Citation: Chen Xiangwei, Zhang Yey, Mei Fengxiang. STABLE GENERALIZED BIRKHOFF SYSTEMS CONSTRUCTED BY USING A GRADIENT SYSTEM WITH NON-SYMMETRICAL NEGATIVE-DEFINITE MATRIX[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(1): 149-153.

## STABLE GENERALIZED BIRKHOFF SYSTEMS CONSTRUCTED BY USING A GRADIENT SYSTEM WITH NON-SYMMETRICAL NEGATIVE-DEFINITE MATRIX

• 摘要: Birkhoff系统是一类比Hamilton系统更广泛的约束力学系统，可在原子与分子物理，强子物理中找到应用.非定常约束力学系统的稳定性研究是重要而又困难的课题，用构造Lyapunov函数的直接方法来研究稳定性问题有很大难度，其中如何构造Lyapunov函数是永远的开放问题.本文给出一种间接方法，即梯度系统方法.提出一类梯度系统，其矩阵是负定非对称的，这类梯度系统的解可以是稳定的或渐近稳定的.梯度系统特别适合用Lyapunov函数来研究，其中的函数V通常取为Lyapunov函数.列出广义Birkhoff系统的运动方程，广义Birkhoff系统是一类广泛约束力学系统.当其中的附加项取为零时，它成为Birkhoff系统，完整约束系统和非完整约束系统都可纳入该系统.给出广义Birkhoff系统的解可以是稳定的或渐近稳定的条件，进一步利用矩阵为负定非对称的梯度系统构造出一些解为稳定或渐近稳定的广义Birkhoff系统.该方法也适合其他约束力学系统.最后用算例说明结果的应用.

Abstract: The Birkhoff system is a more extensive constrained mechanical system than Hamilton system, which can be applied to atomic and molecular physics, and hadron physics.It is an important and difficult project to study the stability of non-steady mechanical system, and it is very difficult to study the stability by using the direct method of constructing Lyapunov function, here how to construct the Lyapunov function is always an open question.This paper gives an indirect method which is called the gradient system method.A kind of gradient systems with non-symmetrical negative-definite matrix is proposed, and the solution of the gradient system can be stable or asymptotic stable.The study of the gradient system is particularly suitable by using the method of Lyapunov functions, in which the function V is usually taken as the Lyapunov function.Firstly the equations of motion of the generalized Birkhoff system are listed.The generalized Birkhoff system is a kind of extensive constrained mechanical system, holonomic and nonholonomic constraint systems can be incorporated into the system.When the additional terms of the system are equal to zero, it becomes the Birkhoff system.Then the conditions under which the solutions of the generalized Birkhoff system can be stable or asymptotically stable are given.Further the generalized Birkhoff systems whose solution is stable are constructed by using the gradient system with non-symmetrical negative-definite matrix.The method is also suitable for the study of other constrained mechanical systems.Lastly some examples are given to illustrate the application of the results.

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