Lie derivative

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so if the vector field is zero at p, it does not mean the flow is contant at p?
actually, yes, it does mean that:

let X be the vector field in question such that X(p)=0.

By definition the flow is the unique path of local diffeomorphims f_t given pointwise in the domain of X by the differential equation (d/dt)(f_t (q))=X(q) when t=0 with initial condition that f_0(q)=q at all points q in the domain of X.

Clearly, if X(p)=0, then f_t(p)=f_0(p)=p.

This is the reason why I kept pointing out that the flow is a path of diffeomorphisms, not just one diffeo. For explain's attempted counterexample, the flow is simply the path of rotations about the origin on the plane, all of which of course fix the origin.
 

mathwonk

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explain seems to be arguing that even though the flow fixes the origin that it does not induce the identity on the tangent space there.
 
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explain seems to be arguing that even though the flow fixes the origin that it does not induce the identity on the tangent space there.
he's right about the action of the flow on the tangent bundle at the origin. but he also says that "there is no flow strictly at the center" which is incorrect.

the flow of the v.f. X=(-y,x) is easy to describe:
f_t(x,y)=((cos t) x - (sin t) y, (sin t) x + (cos t) y).

in fact, calculating flows is kind of fun, try it yourself:

1. In R^3, set X=(1, 0, y). Calculate the flow f_t
2. Set Y=(0, 1, -x). Calculate the flow g_t
3. Calculate [X,Y] and compare that with the path of
diffeomorphisms given by
g_{-t} circ f_{-t} circ g_{t}circ f_{t}
 

mathwonk

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i am still a little behind, but the point i was missing is that a diffeomorphism defines a tangent bundle isomorphism, and at a fix point of the diffeomorphism, the tangent space map can be almost anything, not necessarily the identity.

indeed its behavior reflects the geometry of the vector field and flow near the point. its eigenvalues help describe the geometry of the flow, and in the case of the flow associated to the gradient say of a morse function, even the geometry of the manifold.
 
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explain seems to be arguing that even though the flow fixes the origin that it does not induce the identity on the tangent space there.
having re-read explain's posting, i now understand what he meant by "there is no flow": literally, no 0th order movement at p, just 1st order movement. so, i was wrong in asserting explain's inaccuracy.

your exposition regarding the use of the flows of a vectro field to describe the geometry of a given manifold is exactly right.
 

mathwonk

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explain, this is one of the few times in all the years i have been posting here that i have learned something on a somewhat advanced topic in geometry from a new poster, who clearly understands it better than I do. your post has the authority of someone who understands what he is doing.

you not only are correct in your assertions, but they are elementary and intuitive. moreover you have no axe to grind. welcome to the forum. i think you have a lot to offer.

if you are not yet a mathematician, you can be. indeed you seem to be one
already. if not let me explicitly encourage you.
 
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explain, this is one of the few times in all the years i have been posting here that i have learned something on a somewhat advanced topic in geometry from a new poster, who clearly understands it better than I do. your post has the authority of someone who understands what he is doing.

you not only are correct in your assertions, but they are elementary and intuitive. moreover you have no axe to grind. welcome to the forum. i think you have a lot to offer.

if you are not yet a mathematician, you can be. indeed you seem to be one
already. if not let me explicitly encourage you.
Thank you for your kind words. I am a physicist, but I am not sure I could be a mathematician. I was always frustrated by the textbooks that don't motivate anything but just list a handful of definitions and properties. This is okay when you are learning 2+2=4, but not okay when you are learning the Lie derivative. This frustration of mine is the reason why I always try to find motivation for mathematical notions. Mathematicians seem to be happy just writing lots of definitions and proving lots of statements about structure of some invented objects. There is still a lot of work needed before you can understand what objects are interesting and why. For me, the motivation to study math mainly comes from applications in physics. I am not sure I am sufficiently motivated to study even such "basic" things as algebraic geometry for its own sake.

Indeed, in my post I only meant that the flow does not move at one point, i.e. the center is a fixed point for the flow, but of course the tangent space at that point is mapped nontrivially onto itself.

But I would like to say that for me the Lie derivative meant nothing (I thought it was just an idle mathematician's game) until I saw that one can use it to calculate lots of things in general relativity.
 

mathwonk

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well i used to have a roommate who was a physicist and could explain thigns to me about lie groups and differential geometry, cazimir operators?

anyway i think when a physicist takes it upon himself to understand math concepts he has an advantage because of knowing how they are used and where they arose oftentimes.

riemann apparently used physical intuition in his discoveries and of course ed witten is a physicist who is also one of the best mathematicians.

it is very hard even for a mathematician to understand basic results like stokes theorem unless we try to grasp the idea of a flow, a divergence, a rotation, etc.

i believe these theorems first occurred in the introduction to maxwell's electricity and magnetism.

for me this approach was very hard, not having much physical intuition, and so i made them my own by proving topology results with them, like using green to prove the fundamental theorem of algebra, and gauss to prove there are no never zero vector fields tangent to a asphere.

i later learned (from bott) these are standard arguments in the field of differential topology, but i was just trying to find some motivating ideas to teach from in my several variables class.


also I am a pure mathematician, so to me it was not always acceptable to assume a drop of water is a geometric point, or that the wind moves along a smoothly differentiable curve. to me physics deals with discrete concepts, and it takes a certain knack to reason correctly from erroneous premises!

finally in my old age i realized the mathematical versions of these ideas are merely idealizations of physical notions. but if you learn the ideal notion first it may be harder to grasp the real one.
 
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anyway i think when a physicist takes it upon himself to understand math concepts he has an advantage because of knowing how they are used and where they arose oftentimes.

also I am a pure mathematician, so to me it was not always acceptable to assume a drop of water is a geometric point, or that the wind moves along a smoothly differentiable curve. to me physics deals with discrete concepts, and it takes a certain knack to reason correctly from erroneous premises!
I think it's quite possible to avoid "physical arguments" when dealing with pure mathematics. A student just needs to understand why a new notion is being introduced. For example, in functional analysis: one is supposed to learn a large number of definitions; an operator can be closed, self-adjoint, essentially self-adjoint, bounded, continuous, semi-continuous, upper semi-continuous, weakly upper semi-continuous, closed in strong operator topology, closed in weak operator topology, and so on - ad infinitum. None of this makes any sense beyond a formal agglomeration of abstract constructions, unless a student builds an intuitive picture about why these definitions are useful. This intuitive picture does not need to go outside pure mathematics. The Banach space doesn't have to be the space of solutions of wave equations. But there must be some motivation and intuition. For example, one needs to understand that a given operator (given by some formula) is not always defined on all vectors in an infinite-dimensional space, but an operator can be sometimes extended so that it is defined on more vectors than initially. If an operator can be extended, then it's good to understand how it should be extended, for particular purposes. E.g. to make it self-adjoint or whatever. But if none of this is motivated, then students are left with the impression that this is a difficult and pointless game that requires a perfect photographic memory because you are supposed to memorize 500 definitions and you might use them at any time. I think this destroys motivation for a large portion of math students.

Also, once you go beyond a certain age, your memory is not so good and you <i>need</i> an intuitive picture before you can go on studying a new subject. So there are lots of older professors who never want to learn anything new, because new stuff appears so pointless and incomprehensible. These people probably don't remember that the stuff they learned when they were young also appeared largely pointless and incomprehensible, but they just memorized a large part of it because they could, and the rest somehow made sense. These professors can't teach in any different way either; they just heap their knowledge upon students' heads. This happens in physics as well as in maths.
 

mathwonk

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unfortunately it is difficult to avoid getting older. i have tried with only partial success. Watching spiderman seems to help, or perhaps this is called arrested development.
 
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