General notion of coordinate change

[Logic Cloud]

I'm trying to understand the nature of coordinate transformations in physics. In classical mechanics, we can transform to a different coordinate frame by means of a Galilean transformation. In special relativity, this is replaced by a Lorentz transformation. I am now wondering whether there exists a general notion of coordinate change. More specifically, say we have a function from some manifold M to itself, is there a general set of conditions for this function to qualify as a 'coordinate change'?

(I have encountered the notion of diffeomorphism in the context of general relativity, but I want to confirm whether this is truly the concept I am looking for here.)

 

[Orodruin]

A coordinate change is not a diffeomorphism. It is a mapping from one coordinate chart to another, giving different descriptions of the same manifold. In general you can use whatever coordinates you fancy, orthogonal, curvilinear, etc. They do not need to cover the whole manifold to make a coordinate chart. Coordinate transformations are defined on the overlaps between the coordinate charts.