Velocity Vector has Coordinate-Independent Meaning

In summary: The Levi-Civita connection accounts for this by allowing you to parallel transport along a path while keeping the vector "facing the same way". You can think of parallel transport as a way of keeping track of a vector as you move it around a manifold--if you move it some distance and then parallel transport it back to where you started, you expect to get the same vector. When the manifold is curved you need to specify how to do this, which is where the connection comes in.In summary, the conversation is discussing the concept of the velocity vector and its coordinate-independence in Riemannian Manifolds. It is noted that the velocity vector is an element of the tangent space and can have different components depending on the choice
  • #1
WWGD
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Hi, All:

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:

γ'2(t)=(0,1)

So, in what sense is the velocity coordinate-independent then?

Thanks.

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?

Thanks.
 
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  • #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.

Anyway, Thanks.

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.
 
  • #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.
 
  • #4
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.
 
  • #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.
 

FAQ: Velocity Vector has Coordinate-Independent Meaning

1. What is the significance of velocity vector having coordinate-independent meaning?

The coordinate-independent meaning of a velocity vector is important because it allows us to describe the motion of objects without being dependent on a specific coordinate system. This means that no matter how we choose to measure and describe the motion, the velocity vector will remain the same.

2. How is the coordinate-independent meaning of velocity vector beneficial in scientific research?

The coordinate-independent meaning of velocity vector is beneficial in scientific research because it allows for accurate and consistent measurements of motion regardless of the chosen coordinate system. This reduces the chances of errors and inconsistencies in data analysis.

3. Can you give an example of how the coordinate-independent meaning of velocity vector is applied in real-world situations?

One example is in navigation systems, such as GPS. The velocity vector of a moving object can be determined using satellite data, and this velocity vector remains the same regardless of the chosen coordinate system. This allows for accurate and reliable navigation.

4. How does the coordinate-independent meaning of velocity vector relate to the concept of relative motion?

The coordinate-independent meaning of velocity vector is closely related to relative motion because it allows us to describe the motion of objects in relation to each other without being dependent on a specific coordinate system. This is particularly useful in situations where the reference frame is not fixed or when dealing with moving objects.

5. Are there any limitations to the coordinate-independent meaning of velocity vector?

While the coordinate-independent meaning of velocity vector is useful in many situations, it does have limitations. In some cases, the velocity vector may not accurately represent the motion of an object due to factors such as acceleration, rotation, or changing reference frames. Additionally, the coordinate-independent meaning may not be applicable in non-Euclidean spaces.

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