We're having lectures about Lie Algebras right now, and those were the topics the last time. Our prof is not doing a good job at explaining, he basically reads down his notes without trying to explain the intuition, interpretation and motivation behind those concepts.
I think I get what the Lie Algebra is all about (basically, you have a system of equations/diffeomorphisms (whose IFG are the elements of the Lie Algebra), and using the Lie-bracket operation (so called "commutator"), you can create a group).
But we then went straight into the "Theory of prolongations, Criterion of invariance and splitting of defining equations". And I don't see how it is connected to the knowledge I already have about Lie Algebras.
The only thing I got was that we are now dealing with two systems of equations/diffeomorphisms where:
$$\bar{x}^k=f^k\left(x,u;a\right)=x^k+a\xi ^k\left(x,u\right)+O\left(a^2\right)$$
$$\bar{u}^{\alpha}=g^{\alpha}\left(x,u;a\right)=x^k+a\eta ^k\left(x,u\right)+O\left(a^2\right)$$
Where k=1,...,n and $\alpha$=1,...,m and of course $x\in \mathbb{R}^n$ and $u\in \mathbb{R}^m$
Of course $\xi$ and $\eta$ are the coordinates of the corresponding IFG
This is where I got lost
We then define a manifold $\left(x,u,\partial u,...,\partial ^ru\right)$
This will get us a system of equations:
$$R^{\sigma }\left(x,u,\partial u,...,\partial^ru\right)=0$$
For $\sigma$=1,...,s
We are then creating a one-parameter group $\left\{G_{\alpha }\right\}$ which is the set of those two systems of equations and is admitted to this $R^{\sigma }\left(x,u,\partial u,...,\partial^ru\right)=0$ thing if it maps each solution of that into some other solution of this system
It continues that $R^{\sigma }\left(x,u,\partial u,...,\partial^ru\right)=0$ implies for sufficiently small a that $R^{\sigma }\left(\bar{x},\bar{u},\partial \bar{u},...,\partial^r \bar{u} \right)=0$
Then we start talking about the Galilei Group, we go talking about the Theory of Prolongations and end on the Criterion of invariance & Splitting of defining equations
Because the script is in English I could send it to anyone interested here, it's a longer read but I really don't understand anything of it. I simply don't see the motivation, interpretation and intuition behind all of that, and what it still has got to do with "Solving ODEs with the Symmetry Methods"
Do you maybe know where I could inform myself better about it? Like which book/PDF/Youtube Videos/etc would you recommend?
