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In all literature I have searched, the Galilean transform between two coordinates $(\overrightarrow{x},t)$ and $(\overrightarrow{x'},t')$ have been considered for a "constant velocity".

That is, there exists a constant vector $\overrightarrow{v}$ such that \begin{equation} t'=t \text{ and } \overrightarrow{x'}=\overrightarrow{x}-\overrightarrow{v}t \end{equation} provided that the two coordinate systems share the origin.

Now, I wonder if we can exend the above transformation to cases where $\overrightarrow{v}$ itself is a smooth vector field depending on $\overrightarrow{x}$ and $t$. That is, we have $\overrightarrow{v}=\overrightarrow{v}(\overrightarrow{x},t)$.

Then, how should we make sense of the coordinate system $(\overrightarrow{x'},t')$ that is moving with the velocitiy field $\overrightarrow{v}(\overrightarrow{x},t)$ with respect to the coordinate system $(\overrightarrow{x},t)$?

Since I am remaining in the classical level, I guess it must be like \begin{equation} t'=t \text{ and } x'=x-\int_0^t \overrightarrow{v}(\overrightarrow{x},\tau) d\tau? \end{equation}

But I am not sure... because the velocity field is dependent on the position as well... Could anyone please provide any insght?

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  • $\begingroup$ One would first have to wonder, why? Non-uniform velocities means that this new frame of reference is going to be accelerating quite a lot, and Newtonian mechanics would not be applicable to that. We do use the maths for such a thing, but they are well-motivated in much deeper analysis, not in the case you are looking at. $\endgroup$
    – naturallyInconsistent
    Commented May 10, 2023 at 11:58
  • $\begingroup$ I am just looking for a consistent way to extend to the non-uniform velocity case, which would reduce to the Galilean transform when the velocity is constant. Could you suggest any reference at least? $\endgroup$
    – Keith
    Commented May 10, 2023 at 12:15
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    $\begingroup$ It seems that you are basically asking if one can apply the formalism of general relativity to non-relativistic systems. Yes, one can. I am not aware that it produces all that much interesting physics. It's basically the question what the world looks like from the vantage point of observers who are flowing with some fluid flow and so you will find this applied in fluid mechanics to come extent. $\endgroup$
    – FlatterMann
    Commented May 10, 2023 at 12:16

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