Lie derivation and flows

1. Feb 21, 2009

mma

Lie derivative of a vector field Y along a vector field X is a third vector field acting on a funcion f as
$$\mathcal{L}_X Y(f)(p) = X(Y(f))(p) - Y(X(f)(p) = \lim_{t,s \to 0} \frac{f(\psi_s \circ \phi_t (p)) - f(\phi_t \circ \psi_s (p))}{st}$$

where $$\phi$$ and $$\psi$$ are the flows generated by X and Y respectively.

On the other hand, using an alternative definitiion of the Lie derivative

$$(\mathcal{L}_X Y)_p=\left.\frac{d}{dt}\right|_{t=0}\left((\phi_t^{-1})_*Y_{\phi_{t}(p)}\right)$$

we get

$$\mathcal{L}_X Y(f)(p) = \lim_{t,s \to 0} \frac{f( \phi_t^{-1} \circ \psi_s \circ \phi_t (p)) - f(\psi_s (p))}{st}$$

Are these equal?

2. Feb 21, 2009

rrogers

Since there is no replies yet; I will venture an opinion.
Yes, and I would tackle it using the chain rule for a simple proof. Unfortunately I am lacking reference books right now, so I won't try to write it out.
Moving away from that there are two reasons/models
1) Geometric: Following the drags around the loop they almost measure the same gap. And in the limit would be measuring the gap.
2) Using the Lie derivative implementation in differential geometry one has a straightforward equivalence via. the Riemann tensor.

Sorry for the lack of detail, but I would like to look a couple of things up before being detailed; and I don't have my books. It has also been several years since I studied Lie derivatives.
I imagine that there are C^2 (smoothness) restrictions.

Ray

3. Feb 23, 2009

mma

Hi Ray,

Meanwhile I found the proof in Spivak's book (vol 1., page 155.) The proof is tricky a little bit, but very short.

Cheers,
mma

4. Feb 23, 2009

rrogers

Why not post it, or email it to me. I'm interested in how it would be done by Spivak. I don't have his books.

Ray

5. Feb 23, 2009

mma

This is his proof.

6. Feb 24, 2009

rrogers

Thanks! As usual, in my old age, I have to study Spivak's proofs carefully to see what he has done. In the end it becomes obvious (and "why didn't I do that") though.

Ray

7. Feb 24, 2009

rrogers

mma:

Spivak question: It seems to me that the notation L_x Y in the second to last line is a typo (or I don't understand); it should probably be the partial derivative. The last line is L_x Y by definition?

Ray

8. Feb 24, 2009

mma

To tell you the truth, this proof doesn't satisfy me completely.

I conjecture that the following exrpessions are all equal:

$$\lim_{t,s \to 0} \frac{f(\psi_s^{-1} \circ \phi_t^{-1} \circ \psi_s \circ \phi_t (p)) - f(p)}{st}$$

$$\lim_{t,s \to 0} \frac{f( \phi_t^{-1} \circ \psi_s \circ \phi_t (p)) - f(\psi_s (p))}{st}$$

$$\lim_{t,s \to 0} \frac{f(\psi_s \circ \phi_t (p)) - f(\phi_t \circ \psi_s (p))}{st}$$

$$\lim_{t,s \to 0} \frac{f( \phi_t (p)) - f(\psi_t^{-1} \circ \phi_t \circ \psi_s (p))}{st}$$

$$\lim_{t,s \to 0} \frac{f(p) - f( \phi_t^{-1} \circ \psi_t^{-1} \circ \phi_t \circ \psi_s (p))}{st}$$

but Spvak's proof doesn't give a clue to me for proving this.

9. Feb 24, 2009

rrogers

Looks like a perfect setup for commutation diagram(s) to me.

Ray

10. Feb 24, 2009

mma

I'm afraid that i haven't the vaguest idea what do you mean.
What are these commutation diagrams at all?

11. Feb 25, 2009

rrogers

That's to bad. I was hoping for some help:)
I use them to keep track of mutiple transforms/mappings. Generally they provide a graphical way to illustrate some mappings. I will attempt to repharse your equations in this form. If I make any progress I will post it. Unfortunately I don't have my reference books so I will probably just invent some things. I have forgotten (not that I ever knew thoroughly) how to show anticommutation operators though.
http://en.wikipedia.org/wiki/Commutative_diagram

Ray

12. Mar 1, 2009

mma

It seems that this conjecture is wrong.
Reading further Spivak's book, on page 162 we find a proof for

$$\lim_{t \to 0} \frac{f(\psi_t^{-1} \circ \phi_t^{-1} \circ \psi_t \circ \phi_t (p)) - f(p)}{t^2} = 2[X,Y]f$$

This is just the double of the second and third expression. This is suprising for me a little bit.

13. Mar 1, 2009

rrogers

Was the dropping of the subscript "s" intentional.
In any case, I really have to review my books when they arrive!

Ray

14. Mar 1, 2009

mma

Not dropped, only took equal to t, because Spivak did it also. Strictly speaking, this is really not an exactly equivalent expression, but if the limit on the (s,t) plane exists, then the value of the limit is equal to the the limit above the t=s line.

This proof takes more then one page, and the necessary prerequisits are more then two pages.
The chain rule plays the key role!

Last edited: Mar 1, 2009
15. Mar 2, 2009

mma

Until they arrive...

Last edited: Mar 2, 2009
16. Mar 2, 2009

mma

Sorry, it didn't fit into 3 pieces, here is the end of the proof.

17. Mar 2, 2009

rrogers

Sigh, this is really bad. I have gone from wondering to having doubts about Spivak's reasoning. Do you want to discuss the doubts? In any case, I will look around on the internet for alternate explanations. I really prefer books(:

Ray

18. Mar 2, 2009

mma

Sorry, this post slipped my notice:

No, I think that everything is OK here. $$(L_XYf)(p)$$ means here $$(L_X(Yf))(p)$$. Since $$Yf$$ is a real valued function, $$L_X(Yf)$$ is simply $$X(Yf)$$.
The last line is the definition of $$[X,Y]f$$.

19. Mar 2, 2009

mma

I'm afraid that I don't really know what doubts do you mean.

20. Mar 3, 2009

rrogers

" LaTeX Code: (L_XYf)(p) means here LaTeX Code: (L_X(Yf))(p) "
Thanks, just getting old.
My problems with the second proof:
1) First page
"If there happens to be a coordinate system x"
then
"even if [X,Y]$$\neq$$0 "
While you can build such a coordinate system at p; the coordinate flows will lift
off of the X,Y flow lines at (h,h, 0,0 ). In other words, in general the required
coordinate system doesn't exist?
2)
At the bottom of first page c() is defined as a point. Whereas at the top of the
second page it is a "constant curve". What is a "constant curve"? Then it's
treated as a function. In any case the definition of c as an entity with
p ->0 ends up defining it as a member of T* ; i.e. a path in the total tangent
space. In that case c'() should have to be defined in a coordinate free manner.

It looks to me as if he needs an entity for the calculations and worked backward to
it; then mislabeled it. Of course I consistently misinterpret things at times;
like the original L_x(Yf) above.

Ray