THE CANONICAL HAMILTONIAN REPRESENTATIONS IN A CLASS OF PARTIAL DIFFERENTIAL EQUATION 1)
一类偏微分方程的Hamilton正则表示

The infinite dimensional Hamiltonian system plays a very important role in mechanics, but many partial differential equations that appeared from concrete problems are not in the forms of infinite dimensional Hamiltonian system. Therefore, in recent years, many people are concerned about the problem of which partial differential equation, especially, the linear partial differential equations with variable coefficients and nonlinear partial differential equations can be transformed into the infinite dimensional Hamiltonian system so that the Hamiltonian system is equivalent to the original equations and the introduced variation needed is as few as possible. It has been found that some equations in mathematics, physics, mechanics, have various Hamilton forms . In this paper, based on another definition of infinite dimensional Hamiltonian system, the general method is obtained by using algebra method and operator equation, this method includes criterion principle and concrete infinite dimensional Hamiltonian system of a class of partial differentional equation. In addition, the common method of finding out the Legendre's transformation and the Hamiltonian functional to construct the infinite dimensional Hamiltonian forms is not used in this paper. By using the above method, some canonical Hamiltonian representations are given for the linear differential equations with variable coefficients, KdV equation, MKdV equation, KP equation and Boussinesq equation.