I read in chapter 2 "Weisner Method" in the book "Obtaining Generating Functions"
I did not understand the meaning of this statement
" The method is based on finding a nontrivial continuous group of transformations under which this partial differential equation is invariant. "
I was take a courses in group theory and differential equations in the past, but i did not hear about this .
1- Whats does he mean by continuous group of transformations ?
2- what does he mean by partial differential equation is invariant?
please , anyone can help me ?
Thanks
"Continuous group of transformations" is an old term for Lie group. This is, I think, the terminology that Sophus Lie used, and it was still current in 1933 when Eisenhart published his book Continuous Groups Of Transformations.
To say that a partial differential equation is invariant under the action of a Lie group just means that composition with any group transformation takes solutions to solutions. More explicitly, suppose $G$ is a Lie group acting smoothly on a smooth manifold $M$, meaning there's a map $\theta\colon G\times M\to M$, written $(g,x)\mapsto \theta_g(x)$, which satisfies $\theta_{g_1}\circ\theta_{g_2}=\theta_{g_1g_2}$ and $\theta_{\text{id}}=\text{id}$. If $P\colon C^\infty(M)\to C^\infty(M)$ is a partial differential operator, then the PDE $P(u)=0$ is invariant under the group action if whenever $u$ satisfies $P(u)=0$, then for every $g\in G$, $u\circ \theta_g$ satisfies $P(u\circ\theta_g)=0$. I believe that Lie's main motivation for introducing continuous groups of transformations was to simplify the study of PDEs by exploiting their symmetries, much as the study of an ODE can be simplified if you know it's invariant under rotations or translations.