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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.