Time Parameterizations and Diffeomorphisms

[FreHam]

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.

 

[arkajad]

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.

 

[FreHam]

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.

 

[arkajad]

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.

 

[FreHam]

Great, thanks a lot!