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i was curious as to what exactly this is and more importantly, what motivates it. what are its applications?
Well, as its name suggests, it's a derivative. Directional derivatives were very useful in calculus, weren't they? So you can surely appreciate that a notion of directional derivative would be useful when doing calculus on a manifold.i was curious as to what exactly this is and more importantly, what motivates it. what are its applications?
OMF, this is still misleading IMO. By definition, the Lie derivative takes a vector fields to a new vector field. A vector field takes a function to a new function. Levels of structure are well explained in textbooks like Boothby or Lee, and should not be confused.I should have clarified that the Lie derivative is a second derivative of "functions" on the manifold, not of vectors. It is a first derivate of vector fields, which are themselves first derivatives of functions.
nice site design - but extremely complicated notation...
If vectors are considered to be derivatives, then surely the derivative of a vector is in some sense a second derivative. It's construction is such that it lies in the tangent plane, and therefore is itself "only" a first derivative. But you did have to take a second derivative to get it, so it is in some sense a degenerate second derivative.OMF, this is still misleading IMO. By definition, the Lie derivative takes a vector fields to a new vector field. A vector field takes a function to a new function. Levels of structure are well explained in textbooks like Boothby or Lee, and should not be confused.
The maintainer of that site is undergoing an attempt to fix differential geometry notation, which is admittedly broken. Wish him well.nice site design - but extremely complicated notation...
That's because you entirely missed the point of what Arnold was trying to tell you, which is the same thing I am trying to tell you: yes, prima facie you'd expect [itex][A, \, B][/itex] to give a second order operator, which would prevent it from being a vector field, since vector fields are first order linear homogeneous operators. So it is a surprise that the bracket gives a vector field. IOW, [itex] B \mapsto L_A \, B[/itex] takes a vector field to a vector field, rather than to something more mysterious.As my appeal to authority for the day, In Mathematical Methods of Classical Physics, Arnold originally constructs the Lie Derivative as the second order ,mixed, partial derivative of a function on the manifold. He does later point out that it is a first order operator though, and goes on from this to point out that because of this Lie differentiation as an operator turns vector fields into a group. But what use all this is, I'm not sure.
The fact that the second derivative terms canceled out in the Lie derivative did not escape me. Nevertheless, the fact remains that second derivatives were taken in order to compute it. The higher derivatives canceled to leave only first order terms, which was the whole point of the operation, but nonetheless the derivative can be regarded as one of second order, albiet a degenerate case. If your functions are once but not twice differentiable, you will not be able to obtain the Lie derivative, despite being able to obtain the vector derivative.That's because you entirely missed the point of what Arnold was trying to tell you, which is the same thing I am trying to tell you: yes, prima facie you'd expect [itex][A, \, B][/itex] to give a second order operator, which would prevent it from being a vector field, since vector fields are first order linear homogeneous operators.
Huh? The Lie derivative of a scalar field exists for any once-differentiable function. The Lie derivative of a tangent vector field doesn't involve scalar fields at all. So just what do you mean?If your functions are once but not twice differentiable, you will not be able to obtain the Lie derivative, despite being able to obtain the vector derivative.
It's a directional derivative. You think directional derivatives are pointless? And what is this "opaqueness" of which you speak? And why is it a bad thing, as you seem to imply?If the Lie derivative is simply just another first order vector field, there seems to be little point to it beyond the usual opaqueness.
That's not my understanding from the derivation. As derived aboveHuh? The Lie derivative of a scalar field exists for any once-differentiable function. The Lie derivative of a tangent vector field doesn't involve scalar fields at all. So just what do you mean?
The second derivatives ordinarily cancel. But if the second derivative does not exist, then there is no defined way to cancel the terms. For example letChris Hillman said:[tex] [A, \, B] \, f = A \, B \, f - B \, A \, f = a \, \partial_x \left( b \, f_x \right) - b \, \partial_x
\left(a \, f_x \right) [/tex]
[tex]
\hspace{2cm}
= a \, b_x \, f_x + a \, b \, f_{xx} - b \, a_x \, f_x - a \, b \, f_{xx}
= \left( a \, b_x - b \, a_x \right) f_x
[/tex]
Introducing news ones would be if they are serving no purpose. To be honest I've never come across a differential geometry text that actually uses the Lie derivative as anything other than notational compression, so I'm not sure exactly what use it is in practice. As to the opaqueness, I'm referring to the usual listing of properties, Jacobi identity,etc , which while interesting to know, offer no insight into what this derivative actually does, or what it is used for, which is probably why the original question was asked in the first place.It's a directional derivative. You think directional derivatives are pointless? And what is this "opaqueness" of which you speak? And why is it a bad thing, as you seem to imply?
I completely understand this impression re: Lie derivatives, OMF. The role of Lie derivatives is often understated in the literature. My experience has been that its importance (and it is *very* important) gets passed on more verbally than any other way....To be honest I've never come across a differential geometry text that actually uses the Lie derivative as anything other than notational compression, so I'm not sure exactly what use it is in practice...
Your question seems to be that math books explain how to define the Lie derivative but not why it is defined in this way, and also they do not explain why this construction is useful at all. (It seems that you meant this by the words "usual opaqueness".) It was also my impression that in many books on general relativity people introduce the Lie derivative, differential forms, etc., but then never use them in actual calculations, except maybe using the Lie derivative to formulate the condition for the Killing vector field (but never actually calculating anything with the Lie derivative itself).If the Lie derivative is simply just another first order vector field, there seems to be little point to it beyond the usual opaqueness. I looked at it as a the "best worst choice" for an intrinsic second derivative, because it seemed to serve little other function besides notational convienience. This may be a matter of complete pedantics, but it makes sense to me to regard it as such.
Careful: a flow is a path of local diffeomorphisms....A vector field on a manifold generates a flow (a diffeomorphism) ...
Yes, I agree that strictly speaking this is what one should say. One can follow the flow lines for a finite value of the flow parameter and get a single specific diffeomorphism, so the "entire flow" is a set of many diffeomorphisms.Careful: a flow is a path of local diffeomorphisms.
No, not necessarily. Consider a simple example of a flow: suppose that the flow goes around in circles, with all circles centered at one point. The vector field corresponding to this flow might have components (y, -x) in Cartesian coordinates. Then there is no flow strictly at the center, but there is already a little bit of flow infinitesimally close to the center. Vectors at the center are rotated by the flow.so if the vector field is zero at p, it does not mean the flow is contant at p?