摘要
The idea of the gradient method for integrating the dynamical equations of a nonconservative system presented by Vujanovic is transplanted to a Birkhoffian system, which is a new method for the integration of Birkhoff＇s equations. First, the differential equations of motion of the Birkhoffian system are written out. Secondly, 2n Birkhoff＇s variables are divided into two parts, and assume that a part of the variables is the functions of the remaining part of the variables and time. Thereby, the basic quasi-linear partial differential equations are established. If a complete solution of the basic partial differential equations is sought out, the solution of the problem is given by a set of algebraic equations. Since one can choose n arbitrary Birkhoff＇s variables as the functions of n remains of variables and time in a specific problem, the method has flexibility. The major difficulty of this method lies in finding a complete solution of the basic partial differential equation. Once one finds the complete solution, the motion of the systems can be obtained without doing further integration. Finally, two examples are given to illustrate the application of the results.
将Vujanovi提出的用于积分完整非保守系统动力学方程的梯度法思想移植到Birkhoff系统,给出了Birk-hoff系统积分的一种新方法.首先,列写出Birkhoff系统的运动微分方程;其次,将2n个Birkhoff变量分成2部分,并假设其中一部分变量是其余变量及时间的函数,由此建立拟线性的基本偏微分方程组;如果求出此基本偏微分方程的完全解,则问题的解由一组代数方程给出.该方法具有灵活性,对于具体问题可选Birkhoff变量中任意n个变量作为余下n个变量和时间的函数,其主要困难在于求基本偏微分方程的完全解.一旦求出完全解,就可以不用进一步积分而求得系统的运动.最后,举例说明结果的应用.
基金
The National Natural Science Foundation of China(No.10972151)
