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I am reading Arnold's Mathematical Methods of Classical Mechanics. I have some questions about the definition of world-line. The book says:

A curve in Galilean space which appears in some (and therefore every) Galilean coordinate system as the graph of a motion, is called world-line.

In my opinion, A curve does not necessary appear in every coordinate systems as the graph of a motion just because it does so in some coordinate system. there may be many counter examples. However, I think the proposition is true if galilean coordinate systems are restricted to one-to-one mappings from the Galilean space to $\mathbb{R} \times\ \mathbb{R}^3$ which preserve the Galilean structure.

Am I wrong? Please give me some advice or opinion.

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    $\begingroup$ Arnold is right because when changing reference frame, the generic coordinate transformation is $t'=t+ c$ , $x' = c(t) + R(t) x$ and this transformation transforms a motion into a motion. $\endgroup$
    – Valter Moretti
    Commented Dec 2, 2020 at 19:09
  • $\begingroup$ Thank you for your comment. Why is the coordinate transformation the form? In this book, the definition of coordinate system on a set is merely a one-to-one mapping from the set to $\mathbb{R} \times \mathbb{R}^3$. $\endgroup$
    – Saito
    Commented Dec 3, 2020 at 1:31
  • $\begingroup$ And I believe the transformation is the form if the two coordinate systems are induced by two mappings which preserve the Galilean structure. This is because the transformation is a Galilean transformation and this is why I said coordinate systems should be restricted to mappings preserving the structure in the above question. @ValterMoretti $\endgroup$
    – Saito
    Commented Dec 3, 2020 at 1:47

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