Differential Geometry page 193 Nakahara

[smallgirl]

Hey,

I am struggling to understand what the following is in terms of the mathematics (see Nakahara page 193 at the bottom https://books.google.co.uk/books?id...page&q=non commutativity of two flows&f=false). I understand where the second term in brackets comes from but anything else. So lost...

Thank you for any help.

 

[Drakkith]

Hi smallgirl,

I can't view that particular page. Can you type everything out instead or perhaps post a screenshot?

 

[smallgirl]

https://books.google.co.uk/books?id...page&q=non commutativity of two flows&f=false

Page 193

That should work...

 

[Drakkith]

It does not.

 

[smallgirl]

Geometrically, the Lie bracket shows the non-commutativity of two flows.
This is easily observed from the following consideration. Let $\sigma(s, x)$ and $\tau(t, x)$
be two flows generated by vector fields X and Y , as before, see figure 5.13. If we
move by a small parameter distance ε along the flow σ first, then by δ along τ ,
we shall be at the point whose coordinates are

\begin{eqnarray}
\tau(\delta,sigma(\epsilon,x))&\approx \tau^{\mu}(\delta,x^{\nu}+\epsilon x^{\nu})\\
&\approx x^{\mu}+\epsilon X^{\mu}(x)+\delta Y^{\mu}x^{\nu}+\epsilon x^{\nu}\\
&\approx x^{\mu}+\epsilon X^{\mu}(x) +\delta Y^{\mu}(x)+\epsilon\delta X^{\nu}(x)\partial_{\nu}Y^{\nu}(x)
\end{eqnarray}

 

[Ben Niehoff]

It's not really clear what your question is. Which part of the formula is confusing?

In any case, the picture at the top of page 194 clarifies things a lot.

 

[smallgirl]

The very first line of the formula confuses me. I've not seen notation like this before so I don't understand most of the second line.

 

[Ben Niehoff]

##\delta## and ##\epsilon## are small quantities, and ##\sigma## and ##\tau## are flows. Try backtracking over the previous few pages and find where he defines flows. These formulas at the bottom of page 193 are just flows acting by some infinitesimal amounts (##\delta## and ##\epsilon##).

 

[smallgirl]

I understand that. When I say notation I mean what is the tau doing to the things inside the brackets. Usually when I see bracket notation like ( , ) I think of inner product but that doesn't seem right to me.

 

[fresh_42]

If you go to the definition of a flow you will see it is an exponential series. Since ε and δ are very small, all non linear terms are omitted. That explains the ≈, too.

 

[Ben Niehoff]

I understand that. When I say notation I mean what is the tau doing to the things inside the brackets. Usually when I see bracket notation like ( , ) I think of inner product but that doesn't seem right to me.

Ah, I see. A flow is a function of two quantities: a displacement ##\delta##, and a point on the manifold ##x^\mu##.

 

[WWGD]

I understand that. When I say notation I mean what is the tau doing to the things inside the brackets. Usually when I see bracket notation like ( , ) I think of inner product but that doesn't seem right to me.

Welcome to the notational nightmare of modern mathematics.

 

[fresh_42]

Welcome to the notational nightmare of modern mathematics.

Greetings from Bourbaki
But honestly I doubt that writing page after page with an absolute minimum of formulas, just text wherever you look, has been easier. Remember the hand written letters mathematicians wrote each other in former times. And in Latin! The nightmare starts with everyone denoting stuff a little different and then physicists come along and with them another bunch of notations and ways of simplified calculations of already accepted names.

 

[WWGD]

Greetings from Bourbaki
But honestly I doubt that writing page after page with an absolute minimum of formulas, just text wherever you look, has been easier. Remember the hand written letters mathematicians wrote each other in former times. And in Latin! The nightmare starts with everyone denoting stuff a little different and then physicists come along and with them another bunch of notations and ways of simplified calculations of already accepted names.

I've heard good things about functional notation, but it does not seem very popular in general.

 

[zinq]

"Welcome to the notational nightmare of modern mathematics."

The only real notational nightmare is old-fashioned tensor notation in coordinates with untold subscripts and superscripts to keep track of.

But I digress. The Lie bracket (or just plain "bracket") of two vector fields can be defined in terms of the concept of a vector field as a differential operator.

Don't let the term differential operator scare you. This idea is just that any vector field V can be applied to a real-valued function f on the same space to get a new function, denoted as Vf. At any point x of the space, the function Vf is defined as the directional derivative in the "direction" V(x) of the function f, at x. (The quotes are because if the length ||V(x)|| is not 1, the definition of directional derivative includes multiplication by ||V(x)||.)

The definition of the Lie bracket of two vector fields V and W (on the same space!) is a new vector field called [V, W], obtained — as the difference between applying them in opposite orders:

[V, W]f = V(Wf) - W(Vf).​

This has a useful interpretation in terms of the flows of each of V and W.

The book in question plays fast and loose with the details, as well as having confusing notation. For a better exposition, see the Wikipedia article at https://en.wikipedia.org/wiki/Lie_bracket_of_vector_fields.

(But to see the pages of the quoted book plug this phrase into Google:

non commutativity of two flows Nakahara
​

and the top hit is the page(s) in question.)

 

[mathwonk]

I am learning this topic now from chapter 5 of volume I of Spivak's Comprehensive Intro to differential geometry. He spends most of this chapter making this topic of vector fields, their associated flows, and the lie derivatives along them (of functions, forms, and vector fields), the last of which is the lie bracket, very clear, with most details filled. I recommend it highly.

 

[WWGD]

"Welcome to the notational nightmare of modern mathematics."

The only real notational nightmare is old-fashioned tensor notation in coordinates with untold subscripts and superscripts to keep track of.

But I digress. The Lie bracket (or just plain "bracket") of two vector fields can be defined in terms of the concept of a vector field as a differential operator.

Don't let the term differential operator scare you. This idea is just that any vector field V can be applied to a real-valued function f on the same space to get a new function, denoted as Vf. At any point x of the space, the function Vf is defined as the directional derivative in the "direction" V(x) of the function f, at x. (The quotes are because if the length ||V(x)|| is not 1, the definition of directional derivative includes multiplication by ||V(x)||.)

The definition of the Lie bracket of two vector fields V and W (on the same space!) is a new vector field called [V, W], obtained — as the difference between applying them in opposite orders:

[V, W]f = V(Wf) - W(Vf).​

This has a useful interpretation in terms of the flows of each of V and W.

The book in question plays fast and loose with the details, as well as having confusing notation. For a better exposition, see the Wikipedia article at https://en.wikipedia.org/wiki/Lie_bracket_of_vector_fields.

(But to see the pages of the quoted book plug this phrase into Google:

non commutativity of two flows Nakahara
​

and the top hit is the page(s) in question.)

Google hits are different for different people. They may even vary for different tabs.

 

[lavinia]

Whenever one has a sufficiently smooth vector field,Y, there is an associated "flow". One imagines dropping a leaf in a stream at the point,p, and watching it flow downstream along a curve that passes through p. This curve,c(t) is called the characteristic through p and the velocity of the leaf after a time,t, is the vector##Y_{c(t)}##. If one starts with a domain and watches the flow of all of the points in the domain, then for each time, ##t##, the domain is mapped onto another domain. One can represent this map by a function. ##ψ##, where ##ψ(p,t)## is the point along the characteristic starting at p after an elapsed time,t.

For two vector fields, ##X## and ##Y##, there are two mapping functions, ##φ## and ##ψ##. If one moves along the characteristic through p associated to ##Y## for a small time increment ,##t##, one arrives at the point, ##ψ(p,t)##. Then flowing from this point along the flow associated to ##X## for a time increment,##s## ,one arrives at the point, ##φ(ψ(p,t),s)## . This is what your book is referring to in the calculation that you posted.

If ## f ## is a function then its value at ##φ(ψ(p,t),s)## may be approximated by the first term of the Taylor series expanded from the point, ##ψ(p,t)## along the characteristic associated to ##X## through ##ψ(p,t)## . This is

##f(φ(ψ(p,t),s)) ≈ f(ψ(p,t)) + sX_{ψ(p,t)}.f ##

- Here ##X_{ψ(p,t)}.f ## is the derivative of along the characteristic through ##ψ(p,t)## that is tangent to ##X##.

Next approximate these two terms along the characteristic along ##Y## through p to get

##f(φ(ψ(p,t),s)) ≈ f(ψ(p,0)) + tY_{ψ(p,0)}.f + sX_{ψ(p,0)}.f + stY_{ψ(p,0)}.X_{ψ(p,t)}.f##.

- Here ##f## and ##ψ(p,t)## are functions of ##t## along the characteristic through ##p## that is tangent to ##Y##.

Similary, one approximates ##f(ψ(φ(p,s),t))## to get ##f(φ(p,0)) + tY_{φ(p,0)}.f + sX_{φ(p,0)}.f + stX_{φ(p,0)}.Y_{φ(p,s)}.f##. Note that ##ψ(φ(p,s),t)## and ##φ(ψ(p,t),s)## are generally different points.

The difference between the values of ##f## at these two points is approximately

##st(X_p.(Y.f) - Y_p.(X.f))## .

This is ##st[X,Y]_{p}.f##.

I think that this is what your book means when it says that the difference in the values of the coordinate functions is approximately proportional to the bracket of the two vector fields.

In general, the operation, ##f→X.Y.f## at a point on a manifold is not a tangent vector. If ##x_i## is a local coordinate system and ##Y=Σa_i∂/∂x_i,X=Σb_i∂/∂x_i## then ##X.Y.f=Σ_{i,j}b_j∂a_i/∂x_j∂f/∂x_i+b_ja_i∂^2f/∂x_j∂x_i##. So ##X.Y.f## involves second order partial derivatives.

However, in the operation,##[X.Y].f=X.Y.f−Y.X.f##,the second order terms cancel. One is left with ##Σ_{i,j}(b_j∂a_i/∂x_j−a_j∂b_i/∂x_j)∂f/∂x_i## so##[X,Y]## is the tangent vector, ##Σ_{i,j}(b_j∂a_i/∂x_j−a_j∂b_i/∂x_j)∂/∂x_i##(Note that one does not need to check if this expression transforms consistently with a change of coordinates because it is already defined intrinsically in terms of directional derivatives with respect to X and Y.)

One can also see from this formula that ##[X,Y]## varies smoothly from point to point and so is a vector field.* A well written reference for fundamentals of ordinary differential equations is Lectures on Ordinary Differential Equations by Hurewicz. Here is a link to the PDF file.

http://www.staff.science.uu.nl/~caval101/homepage/Differential_geometry_2011_files/Hurewicz.pdf )