论文标题
用于常规微分方程的修改正式拉格朗日配方
A modified formal Lagrangian formulation for general differential equations
论文作者
论文摘要
在本文中，我们通过引入依赖变量并证明对任何微分方程系统的公式存在，提出了一种修改的正式拉格朗日公式。相应的Euler-拉格朗日方程由原始系统及其围绕虚拟变量组成的伴随系统，通过简单的替换为虚拟变量，将其简化为原始系统。该公式用于通过Noether定理研究微分方程的保护定律，尤其是Fornberg的非平凡保护定律 - Whitham方程是通过使用其Lie Point对称性获得的。最后，显示了不可压缩的Euler方程的保护定律与相关修改正式Lagrangian的变异对称性之间的对应关系。
In this paper, we propose a modified formal Lagrangian formulation by introducing dummy dependent variables and prove the existence of such a formulation for any system of differential equations. The corresponding Euler--Lagrange equations, consisting of the original system and its adjoint system about the dummy variables, reduce to the original system via a simple substitution for the dummy variables. The formulation is applied to study conservation laws of differential equations through Noether's Theorem and in particular, a nontrivial conservation law of the Fornberg--Whitham equation is obtained by using its Lie point symmetries. Finally, a correspondence between conservation laws of the incompressible Euler equations and variational symmetries of the relevant modified formal Lagrangian is shown.
