一类微分方程的Hamilton系统的正则化

The regularization of the class of differential equations in the Hamilton system

文章以Hamilton系统作为研究背景,探讨了获得新的无穷维Hamilton正则形式的过程.通过带余除法,将一类偏微分方程,转化到无穷维Hamilton线性正则系统下,在此过程中,为了能够整除,令余式为零,从而归纳出获得Hamilton正则形式的操作步骤,并针对某些方程在Hamilton算子为二阶算子时无法获得正则形式的情况,尝试了由二阶升四阶的处理方式,特别讨论了当Hamilton算子为准对角形式时,相应算子所满足的方程组以及具体的Hamilton正则表示形式,在一定程度上实现了获得新的无穷维Hamilton正则形式的机械程序化.该方法的优势在于简单、易操作,同时,为Hamilton算子的获得提供了一条新思路.

In the paper,based on the research background of Hamiltonian system,the process of obtaining a new infinite dimensional Hamiltonian canonical form is discussed.Through division,a class of partial differential equations is transformed into an infinite dimensional Hamiltonian linear canonical system.In this process,in order to be divided with the remainder zero,the operation steps are summarized to obtain the Hamiltonian canonical form.Besides,as for the situation that canonical form can’t be obtained when the Hamiltonian operator of some equations are a second order operator.Thus,the second order was increased to the fourth order with attempts.In particular,the equations satisfied by the corresponding operator and the specific Hamiltonian canonical form was studied when the operator is in quasi diagonal form,which to a certain extent realizes the mechanical programming of obtaining the new infinite Hamiltonian canonical form.The advantage of this method is simple and easy to operate,providing a new idea for obtaining Hamiltonian operator.