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

Stephen Tashi
Hey Stephen!

Have you abandoned this stuff? Or maybe got bored/lonely being here by yourself for too long?
No - at least I tell myself that I'm going to proceed. I'm a retired guy and I have hundred other unfinished projects that distract me. What I need is what everyone needs - motivation! So thanks for the post.

What I'll do next in this thread is give a version of post #67 from the thread https://www.physicsforums.com/showthread.php?t=699669&highlight=infinitesimal&page=4 since it's relevant. In the meantime, if you have any thoughts about post #67, let me know.

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bigfooted
Gold Member
I also don't have the book of Emanuel, but I have the book of Hydon and Stephani and bluman/kumei. I also have the book of Cantwell, since my original interest was fluid dynamics applications. Up to now, the 'meditations' were very interesting and I actually started reading those books again to follow this discussion

EDIT: we want more! we want more! :-)

Looking back I guess my exposition of Cohen was too advanced, all I want to do is be able to solve first & second order ODE's & then maybe PDE's, but unfortunately we did see why you have to be careful with your notation in this & the other thread so there is merit to what I've done, but **** that...

I don't care about the formal theory right now. I'd love to see how you solve separable, euler-homogeneous & linear first order ode's in a way that makes sense of everything you do in lie theory, in a way that motivates the stuff in the early chapters & explains their necessity. I felt I had the basics of the theory down a while ago, but got sidetracked. I'll do my best, but

What I'll do next in this thread is give a version of post #67 from the thread https://www.physicsforums.com/showthread.php?t=699669&highlight=infinitesimal&page=4 since it's relevant. In the meantime, if you have any thoughts about post #67, let me know.
Do you understand Taylor's theorem now my use of Taylor's theorem in that thread? If you don't & can't get access to chapter 14.10 of Thomas calculus to see a nice easy proof I'll copy the pages onto here for you if you need them.

Stephen Tashi
Do you understand Taylor's theorem now my use of Taylor's theorem in that thread?
I understand Taylor's theorem. I don't understand your posts in that thread.

I offer this dichotomy about the Taylor's expansion that uses the operator $U$. One of the following must be true:

1) The expansion expresses a fact that can be stated and confirmed by ordinary calculus. (I'm taliking about modern calculus, not the old-time calculus with infinitesimals.)

2) The expansion expresses a fact that cannot be stated and confirmed by ordinary calculus. It requires more advanced ideas from differential geometry or some other advanced field of mathematics.

I'm investigating 1). As to explanations that take the viewpoint 2), I'm not interested in them at the moment. I'll resort to viewpoint 2), if I don't make progress with 1).

I have yet to see any book give a clear definition of $U$ from the viewpoint of 1) and I have not see any proof of the expansion from the viewpoint of 1). There are certainly "hand-waving" statements that claim the expansion follows from Taylor's theorem, but they are inadequate. Taking the viewpoint 1), it would be necessary to define $U$ in terms of ordinary calculus and no book seems capable of doing that!

First off, going by a quote in this thread, "The exponential map is a local diffeomorphism at the origin, so Taylor's theorem for multivariate functions applies", I'm pretty sure the general exponential map is derived from Taylor's theorem for multivariable functions in the general case anyway, & I can roughly see how that makes sense, thus I think this is the first objection to what you've just said. 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.

Second there's nothing wrong with the derivation of the exponential map given in Taylor's theorem. All you're doing is deriving the taylor expansion of a function, then defining a notation. The general definition in the wikipedia link defined the exponential map as a map on tangent spaces, this general definition applicable to any manifold is nothing but a rigorous way of saying the exact same thing, you've simply grounded the domain in which that operator lives so you can apply it on more general manifolds than ℝ2 (say). The $\tfrac{\partial}{\partial x}dx + \tfrac{\partial}{\partial x}dy$ is a tangent vector in the tangent space, i.e. $\vec{v} = \tfrac{\partial}{\partial x}dx + \tfrac{\partial}{\partial x}dy$ (following the wiki definition section).

Third there seems to be an issue about differentiating $dx$ & $dy$, as I referenced here & showed you guys were doing it due to sloppy notation. If you don't believe my derivations in that thread then by all means learn what a tangent space to a manifold is, & see how the $dx$ & $dy$ in $\vec{v} = dx\tfrac{\partial}{\partial x} + dy \tfrac{\partial}{\partial x}$ are scalars, where $\tfrac{\partial}{\partial x}$ & $\tfrac{\partial}{\partial y}$ are basis vectors in the tangent space, so by the basic formalism of tangent spaces you see it can't make sense to differentiate those as you guys did in that link I just gave since you're plugging them into the exponential map, the important thing in the expansion will be the basis vectors & the scalars merely come along for the ride... Again though, you only need to follow the elementary Taylor's theorem to see this, doing things on a tangent space merely formalizes into definitions (in terms of maps) what we're doing by basic intuition in ℝ2. So follow the derivation very closely again, and refer to my posts in that thread to make sure you follow my explanation of the tiny flaw in what you guys were doing (post 52 applies Taylor's theorem to our specific case involving one-parameter groups).

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.

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