微分方程的Hamilton化与解法

HAMILTONIAN FORMULARIZATION OF DIFFERENTIAL EQUATIONS AND THEIR METHOD OF SOLUTION

提出一种求解微分方程的力学方法.首先,将一类常微分方程化成一个Hamilton方程,在特殊情况下化成Hamilton原来的方程,在一般情况下化成带非保守力的Hamilton方程.其次,利用Hamilton系统的Noether理论求守恒量.如果找到足够多的守恒量,便找到了方程的解.最后,举例说明结果的应用.

A kind of mechanical method for solving differential equations is given. First, a set of differential equation can be written in the form of the Hamilton equations. In particular case, the Hamilton equations possess a canonical form. In general case, they possess a canonical form with non-conservative forces. Secondly, the first integrals of the equations can be obtained by using the Noether theory of the Hamilton system. If all of the integrals can be found, then the solution of the equations will be obtained. Finally, an example is given to illustrate the application of the result.