Need help qualitatively understanding the concept of a tangent bundle

[mjordan2nd]

Hello. In my textbook by Jose Saletan called Classical Dynamics: A Contemporary Approach the author talks about TQ, the domain of the Lagrangian. He states that the space tangent to a point on the configuration manifold is in the tangent bundle, and that the entire tangent bundle can be thought of as just this applied to all points on the configuration manifold (if I'm understanding what he's saying correctly). He also states that the tangent bundle is where the veloicities of the system lie.

He then goes on to give a concrete example: where the configuration manifold is a circle. He states that the tangent bundle then will be a cylinder. I don't really understand why this is. Clearly all the velocity vectors will lie in the plane of the circle. Why should the tangent bundle have components perpendicular to the circle?

 

[tiny-tim]

hello mjordan2nd!

He then goes on to give a concrete example: where the configuration manifold is a circle. He states that the tangent bundle then will be a cylinder. I don't really understand why this is. Clearly all the velocity vectors will lie in the plane of the circle. Why should the tangent bundle have components perpendicular to the circle?

the tangent bundle to a manifold will not generally be drawable conveniently (or at all) in the original space

you should not think of it as part of the original space

the tangent bundle of the circle is the tangent lines at each point, laid side-by-side …

that makes a cylinder, doesn't it? ​

 

[truffaldino]

Just take this formally: Consider the pair (coordinate, velocity). In case of a particle on a circle your configuration space is one dimensional circle with angular coordinate phi. Let the particle velocity be omega. In this case you just have a pair of numbers (phi, omega) where pair (phi+2pi, omega) is equivalent to the pair (phi, omega). This is exactly the way to parametrize cylinder. So your tangent bundle is a cylinder.Edit: (oops: somebody already posted the answer)

 

[mjordan2nd]

I guess the problem I'm having with this is that, from what I understand, in the book the tangent bundle was presented as a space in which the velocity vectors could live. If we think of the circle as embedded in R2, and a particle's motion as being constrained to that circle, then the velocity vector would always be in the plane of the circle. So at point (R,0) the velocity vector would be (0, v1), at point (0,R) the velocity vector would be (v2,0) and so on. So it seems to me that the space in which the velocities lie should be coplanar to the circle. So I'm having a hard time seeing why the tangent bundle should be a cylinder rather than a bunch of coplanar tangent lines extending from the circle. My suspicion is I am misunderstanding the definition of the tangent bundle, but I'm not really sure how/why.

 

[tiny-tim]

hello mjordan2nd!

… a space in which the velocity vectors could live. If we think of the circle as embedded in R2, and a particle's motion as being constrained to that circle, then the velocity vector would always be in the plane of the circle …

consider the tangent bundle of the surface of a sphere …

that's a plane at each point of the sphere …

total 4 dimensions: how can you draw that in 3D ?

i repeat, the tangent bundle has nothing to do with the original space ​

 

[truffaldino]

Do not think about embedding, the tangent bundle just a way to parametize positions and velocities In case of a circle these are just two numbers, one of which is a periodic coordinate.

 

[mjordan2nd]

I see. So what you guys are essentially saying is that the tangent bundle doesn't really have anything to do with how the velocity vectors "look" in the physics I sense of the word, it's just a way to parameterize the position/velocity. And the structure of that space is more dependent on what type of coordinates we're using (in this case, the periodic coordinates give us a cylinder) rather than what the velocity vectors do in R3, and the fact that the tangent bundle is not coplanar with the cylinder does not mean that the velocity vectors are not coplanar with the cylinder. Is this correct?

 

[truffaldino]

I see. So what you guys are essentially saying is that the tangent bundle doesn't really have anything to do with how the velocity vectors "look" in the physics I sense of the word, it's just a way to parameterize the position/velocity. And the structure of that space is more dependent on what type of coordinates we're using (in this case, the periodic coordinates give us a cylinder) rather than what the velocity vectors do in R3, and the fact that the tangent bundle is not coplanar with the cylinder does not mean that the velocity vectors are not coplanar with the cylinder. Is this correct?

It is correct, except dependence of structure on type of coordinates: I would say dependence of structure of TB on structure of configuration space.

 

[mjordan2nd]

Thank you both for your kind replies!