基于Frobenius定理的Hamilton-Jacobi方法的几何解释

A Geometric Explanation of Hamilton-Jacobi Methods Based on the Frobenius Theorem

给出了一阶偏微分方程特征微分方程组的一种基于Frobenius定理的几何解释,通过研究发现根据Frobenius定理可以从一阶偏微分方程直接得到其特征微分方程组;在此基础上说明如何利用几何方法从Hamilton正则方程出发找到与之对应的Hamilton-Jacobi方程.这种方法可以被用于非保守或非完整Hamilton力学问题的研究中,经典Hamilton-Jacobi方法是这种方法的一个特例.

With the differential geometry method, a geometric explanation based on the Frobe- nius theorem for characteristic equations of 1st-order partial differential equations was presen- ted. According to the Frobenius theorem, the characteristic equations can be deduced directly from the 1st-order partial differential equations. Based on this, how to use the geometric meth- od to find the corresponding Hamilton-Jacobi equations from Hamiltonian canonical equations was discussed. This method could be utilized to address the nonconservative or nonholonomic Hamiltonian mechanical problems. The classical Hamilton-Jacobi method is only a special case of this method.