Even knowing some basics of maths and physics I get puzzled when I try to systematise some concepts for better understanding. One is basically on how all the mathematical concepts comprise the model for the space in which classical (non-relativistic) mechanics takes place. Here I am just referring to the typical 3D Euclidean affine space. My specific question is:
What would be the 'staircase' or followup of concepts that needs to be defined in order to later define the Euclidean affine space in which the mechanic takes place (our 'regular' physical space)?
My tentative answer would be as follows:
- vector (or linear) space
- as above with dot product
- metric space
- affine space
- finally Euclidean affine space
More specific questions are:
We usually define (standard) dot product as something like $\sum a_i b_i$. But for our 'regular' space this works only in Cartesian frame. However, at the beginning we define just points, vectors, their norms (without concepts of orthogonality etc.) and distances (metrics), all of them do not need any coordinate system in fact. I find it puzzling we usually already use Cartesian frame at the early stage.
Where does concepts of differentiation etc. come in? Do we need to think of Banach space in fact here? In mechanics some bodies can move, we introduce curves, tangent vectors etc.
I fully understand this question my be a bit broad, but hoping some will have some time to discuss these issues in details. In addition, I would be very pleased to have some literature references in which I can look something like that. Thank in advance.