Using Lie Groups to Solve & Understand First Order ODE's

[Stephen Tashi]

In other words it does indeed seem the general theorem can be confirmed by basic calculus. The wikipedia definition section for the exponential map seems to confirm this.

I find no demonstration of the general result by ordinary calculus in that article. I don't even find a precise statement of the result.

I think a result for matrix groups can be shown by ordinary calculus because matrix groups are special. In a matrix group $x1 = T_x(x,y,\alpha))$ is a linear function in $x$ and $y$. Hence higher order partial derivatives of $x_1$ with respect to $x$ or $y$ vanish.

Problem 2.2 in Emmanuel p 17 uses the example of the group defined by
$x_1 = (x^2 + \alpha x y)^{1/2}$
$y_1 = \frac{xy}{(x^2 + \alpha x y)^{1/2}}$

It would interesting to discuss the specifics of expanding a function of $(x_1,y_1)$ in Taylor series.

It won't do any good to rant at me about how the result is obvious from differential geometry. As I said, I'm perfectly willing to consider that the Taylor expansion in terms of $U$ can only be defined and proven by using concepts from differential geometry. However, for the time being, I'm interested in what can be done with ordinary calculus. People with other interests should feel free to post about them.

 

[bolbteppa]

I think this is exactly the book I wanted to read back in 2013!

https://books.google.ie/books?id=MqcXBgAAQBAJ&dq=arrigo+symmetry&source=gbs_navlinks_s

Thoughts?

 

[ibkev]

I think this is exactly the book I wanted to read back in 2013!
https://books.google.ie/books?id=MqcXBgAAQBAJ&dq=arrigo+symmetry&source=gbs_navlinks_s

Funny you mention that, I also came here to see what people thought of Arrigo's "Symmetry Analysis of Differential Equations: An Introduction". Based on the preview, at amazon it seems quite accessible and his solutions to problems are easy to follow.
https://www.amazon.com/dp/1118721403/?tag=pfamazon01-20