Time Parameterizations and Diffeomorphisms

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Discussion Overview

The discussion revolves around the concept of time parameterizations and diffeomorphisms in the context of dynamical systems defined on a manifold. Participants explore the implications of reparameterizing time in Newtonian systems and how it affects the representation of coordinates on the manifold.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant, Fred, inquires about the nature of the map resulting from a time reparameterization and whether it constitutes a diffeomorphism from M x R to M x R.
  • Another participant clarifies that the diffeomorphism can be expressed as (x,t) -> (x,s(t)), indicating that the x coordinates remain unchanged during the reparameterization.
  • A third participant discusses the context of a Lagrangian system and suggests that the diffeomorphism is a map from TQ x R to TQ x R, expressing a potential understanding of the topic.
  • Further contributions note that the implications of reparameterization depend on whether one is in the tangent bundle or the "bundle of jets," with the latter being less commonly addressed in textbooks.
  • Participants express varying levels of confidence in their understanding, with some indicating a need for further clarification on specific aspects.

Areas of Agreement / Disagreement

Participants express differing views on the implications of time reparameterization, particularly regarding its effects on different mathematical structures (tangent bundles vs. bundles of jets). The discussion remains unresolved with multiple competing perspectives on the topic.

Contextual Notes

Limitations include the potential confusion arising from the different mathematical frameworks being discussed, such as tangent bundles and bundles of jets, which may not be familiar to all participants.

FreHam
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Hello!

It's my first post here, as I am currently reading some material, but have not been able to really grasp it. Sorry, if this is a rather dumb question.

I have a dynamical system (Newtonian) that is defined on some manifold M times R (time-dependent system). Say that time is labeled t. If I now transform the time t\mapsto t(s), which is a map from R to R.

If M has coordinates (x(t),y(t),z(t),u(t),v(t),w(t)) and I want to transform it with the time re-parameterization I get (x(t(s)),y(t(s)),z(t(s)),...). Is that a diffeomorphism from M x R to M x R, or how should I write that properly? Is the real map here not x(t) -> x(t(s)), etc. or does it suffice to say that the full map is just t -> t(s)?

I'm stuck and I have no idea. Again, I apologize for my stupidity, but any help is much appreciated. I could ask my professor, but having asked several things already without receiving much help, I hope you guys (and gals?) might have an answer...

Cheers,

Fred.
 
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If your manifold is M x R and if you reparameterize time, then your manifold is still M x R and the diffeomorphism is described by (x,t) -> (x,s(t)). Reparameterization of time does nothing to the x coordinates. But if you consider maps from R to M (say, sections of the product bundle), then the coordinate description of these sections will change the way you wrote.
 
Yes, it's basically a Lagrangian system, so I have a tangent bundle, M=TQ, where Q is the base manifold with coordinates q, and TQ has coordinates (q,\dot{q}). For a time-dependent Lagrangian L:TQ x R -> R, the diffeomorphism is a map TQ x R -> TQ x R, right? If so, I think I might finally get the hang of it. Otherwise, I'm back to square one.
 
Well, it all depends on the details. Sometimes you are in the tangent bundle and sometimes in the "bundle of of jets" of maps from R to Q. When you re-parametrize time nothing happens to the tangent bundle. But something happens to the "bundle of jets". Jets are not very popular, therefore most textbooks avoid them. But then the readers may get confused as you are.

Things get however simpler when you get to particular problems and applications. Then you know what to calculate and how.
 
Great, thanks a lot!
 

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