|Mar29-12, 07:56 PM||#1|
Velocity Vector has Coordinate-Independent Meaning
It's been a long time since I did this, and I have some basic doubts; please bear with me:
In Lee's Riemannian Mflds, p.48, he states that , given a parametrization :
γ:(a,b)→M, of a curve , "the velocity vector γ'(t) has a coordinate-independent meaning
for each t in M" (this should be for each t in (a,b).
Now, Lee goes on to give an example of two parametrizations of S1 , one
of which is γ(t)=(cost,sint), and the other is the polar-coordinate expression:
γ 2(t)=(r(t),θ(t))=(1,t) .
Now, in the first parametrization, the velocity is given by:
γ'(t)=(-sint, cost) , while
in the second one, we get:
So, in what sense is the velocity coordinate-independent then?
EDIT: Moreover, the reason given for the "ambiguity" in defining acceleration seems to apply to the definition
of velocity too:
In the difference quotient LimΔt→0 [f(x+Δt)-f(x)]/Δt
the vectors x and x+Δt live in tangent spaces that are not naturally isomorphic
to each other, right?
Lastly --hope this is not too long of a question-- I understand at an informal level
that a connection is a device used to define/select a choice of isomorphism between
vector spaces that are not naturally-isomorphic to each other, but I do not see anywhere
in this chapter where/how those isomorphisms are defined. Any Ideas/Suggestions?
|Mar29-12, 08:10 PM||#2|
Moreover, it would be great if someone could guide me thru the case of the circle itself,
on how to define a connection on it. I usually learn more from working with coordinates
and then doing the abstracting myself, than from seeing the material abstracted by
someone else without my knowing what is being abstracted.
Just to note that I read PRof. Quasar and Others' recent post on Connections, but it was stated there that
the post applied to principal bundles, and not to vector bundles.
|Apr1-12, 01:18 PM||#3|
I'm just learning this stuff myself, so please correct me if I'm wrong.
The velocity vector is a coordinate-independent object that is an element of the tangent space of the point you're considering. But like all vector spaces, there is not a unique choice of basis--one usually picks the basis induced by the coordinate lines, ie the partial derivative operators with respect to those coordinates, although you can modify them however you want. So although you derived two different sets of components for the velocity vector, you have (probably--I didn't check your math) derived the same velocity vector. Components specify the weights put on each basis vector so the same vector will generally have different components in a different basis. What always confused me is when people used "change of coordinates" and "change of basis" interchangeably. Really it's the change of basis that matters, and the basis we're talking about is the basis of the tangent space. A change of coordinates will induce a change of basis (if we want it to), but we can consider the basis induced by a particular coordinate system while using a different coordinate system to label points.
|Apr2-12, 05:44 PM||#4|
Velocity Vector has Coordinate-Independent Meaning
Ah, good point. I'll try as an exercise to double-check to see if there is an actual
(covariant) change-of-basis taking us from (-sint,cost), to (0,1). Still, I have not figured
out the second issue: how is it that these connections allow us to overcome the issue
of vector spaces at different points (say, when we have a vector field along a curve)
where the tangent spaces are not naturally isomorphic at each other.
I know in R^n ( by def. I think) vector fields in different tangent spaces are parallel
iff they have the same component. Once we start with moving frames , e.g., Frenet-
Serret frames, then the basis frames are no longer parallel , then their displacement
has to be taken into account. I'll repost when I figure it out.
|Apr2-12, 06:04 PM||#5|
A connection is the same as specifying a rule for parallel transport. You can specify that rule in some slightly arbitrary way on your manifold (the arbitrariness is limited by a few properties that a connection must have).
You could, for example, choose to embed your manifold in a higher dimensional flat manifold (e.g. 2 sphere in R^3) and parallel transport by first parallel transporting in the flat manifold as usual and then projecting the resulting vector (assuming you have a metric) onto your embedded submanifold. The result of this construction is the so-called Levi-Civita connection (one can show that this definition is independent of choice of embedding).
The difference between the flat-manifold case is that on a curved manifold, whether 2 vectors are parallel or not depends on the path through which you connect them.
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