Mathematical model for Euclidean space in classical mechanics 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.
 A: The natural approach consists of starting by assuming that the physical space is an affine $3$-dimensional space
$(\mathbb{A}^3,-, V^3)$
($- : \mathbb A^3 \times \mathbb A^3 \to V^3$ is the map associating couples of points $P,Q$ with vectors $P-Q =: \vec{QP}$)
whose vector space of translations $V^3$ is further equipped with a positive symmetric inner product ${\bf u}\cdot {\bf v}$ for ${\bf u}, {\bf v} \in V^3$.
By definition a positive, symmetric, scalar product is a map
$$ V^3 \times V^3 \ni ({\bf u},{\bf v}) \mapsto  {\bf u}\cdot {\bf v}\in \mathbb{R}$$ that is
(a) linear in each entry,
(b) symmetric: ${\bf u}\cdot {\bf v}={\bf v}\cdot {\bf u}$,
(c) positive: ${\bf u}\cdot {\bf u} \geq 0$ and ${\bf u}\cdot {\bf u}=0$ implies ${\bf u}= {\bf 0}$.
As a consequence,the Euclidean space $$\mathbb{E}^3 \equiv (\mathbb{A}^3,-, V^3, \cdot)$$ becomes a complete metric space whose distance
$$d(P,Q):= \sqrt{(P-Q)\cdot (P-Q)}$$
is translationally invariant:
$$d(P+{\bf u},Q+{\bf u}) = d(P,Q)\:,$$
where $P':= P+{\bf u}$ is the only pont in $\mathbb{A}^3$ such that $P'-P = {\bf u}$.
$V^3$ equipped with the natural norm arising from $\cdot$ turns out to be a Banach space (quite trivially because it is finite dimensional).
With this structure all the differential vector calculus can be developed. For instance, the velocity of a curve $P=P(t)\in \mathbb{A}^3$ is nothing but
$${\bf v}(t) := \frac{dP(t)}{dt} := \lim_{h\to 0} \frac{1}{h}(P(t+h)-P(t))\:.$$
(Actually one only needs an orientation if the vector product is assumed to produce proper elements of $V^3$.)
A $C^\infty$ (actually real analytic) manifold structure can be now introduced by assuming that it is generated by the atlas of Cartesian coordinate systems constructed as follows. Take an origin $O\in \mathbb A^3$ and a basis ${\bf e}_1, {\bf e}_2, {\bf e}_3 \in V^3$ and the (global) associated coordinate system is
$$\mathbb{A}^3 \ni P \mapsto (x^1,x^2,x^3) \in \mathbb{R}^3$$
such that $$P-O= \sum_{k=1}^3 x^k {\bf e}_k\:.$$
One sees that this map is well-defined from the axioms of affine space,  bijective and that different choices of the origin and the basis produce different global charts which are pairwise $C^\infty$ (real analytic) compatible (since the transition functions are non-singular linear generally non-homogeneous transformations).
As an elementray result of linear algebra,(in any dimension actually)  every positive symmetric scalar product admits basis  ${\bf e}_1, {\bf e}_2, {\bf e}_3 \in V^3$
which are orthonormal, namely,
$${\bf e}_i \cdot {\bf e}_j = \delta_{ij}\:,$$
Using these bases, the associated Cartesian coordinate systems are called   orthonormal Cartesian system of coordinates.
Notice that if the basis is in fact chosen as orthonormal then the scalar product, by definition, is
$${\bf u}\cdot {\bf v}= \sum_{j=1}^3 u^jv^j$$
where ${\bf u}= \sum_{j=1}^3 u^j {\bf e}_j$ and ${\bf v}= \sum_{j=1}^3 v^j {\bf e}_j$. Similarly, if the coordinates of $P$ and $Q$ are $x^j(P)$ and $x^j(Q)$ always referring to an orthonormal Cartesian coordinate system, then
we have
$$d(P,Q) = \sqrt{\sum_{j=1}^n (x^j(P)-x^j(Q))^2}\:.$$
As a byproduct of the existence of Cartesian coordinate systems, the natural topology of $\mathbb{A}^3$ is homeomorphically identified with the standard topology of $\mathbb{R}^3$, independently of the Cartesian coordinate system.
Moreover, there exists a canonical isomorphism between the tangent space $T_O\mathbb{E}^3$, for every given $O\in \mathbb{A}^3$,  and $V^3$ taking the form
$$\psi_{O,{\bf e}_1, {\bf e}_2, {\bf e}_3} : \sum_{k=1}^3 t^k \left.\frac{\partial}{\partial x^k}\right|_{O} \mapsto \sum_{k=1}^3 t^k {\bf e}_k$$
where $x^1,x^2,x^3$ are Cartesian coordinates constructed out of $O$ and ${\bf e}_1, {\bf e}_2, {\bf e}_3$. The fact that the isomorphism is canonical just means  that it does not depend on the said choices: $\psi_{O,{\bf e}_1, {\bf e}_2, {\bf e}_3}=\psi_{O',{\bf e}'_1, {\bf e}'_2, {\bf e}'_3}$.
Relying upon this isomorphism all vector (and tensor) analysis can be indifferently developed in $\mathbb{R}^3$ or in $\mathbb{E}^3$ giving rise to isomorphic results.
Finally, $(\mathbb{E}^3, g)$ turns out to be a globally flat $C^\infty$ Riemannian manifold when the metric $g$ is defined as
$$g_O\left({\bf u}, {\bf v}\right) := {\bf u}\cdot {\bf v}$$
for ${\bf u}, {\bf v} \in T_O\mathbb{E}^3$, taking advantage of the aformentioned canonical isomorphism.
(I think I summarized something like 2300 years of mathematical ideas about the physical space.)
An interestig fact is that the Minkowski spacetime has quite the  same structures but the dimension is $4$ and the scalar product is of Lorentz type.
What is different is the relation between metric structures and the underpinning topology which is not metric (with respect to the metric structures since the scalar product is not positive).
Finally all that is mathematics, there is no guarantee for the validity of all the description concerning  our physical space. Indeed, we already know that the affine structure does not exist at least at large scales. At Planck scales even the topological structure may be different.
ADDENDUM Empirical facts at human scales.
How to relate the above construction with geometrical physics?
First of all there is, at human scales, the empirical evidence that rigid bodies exist. In principle, these bodies can be moved around in the space but their dimensions do not change. In particular, we can construct (ideally speaking) a class of rigid rulers corresponding to physical segmentes or physical vectors.
The fact that they are rigid can be checked as follows. Take two such rulers in the class, compare them directly at rest in a place of the space: their endpoints coincide. Next move them separately around in the universe and finally compare them at rest in (another) place of the universe. Their endpoints again coincide.
The existence of physical bodies that satisfy this physical fact (with a number of approximations and corresponding issues) is a physical fact. We are so familar with it that we consider it necessary, but it is not.
Equipped with this class of ideal rigid rulers we can, on the one hand,  measure everything, on the other hand  define straight lines through a practical implementation of the parallel transport. And this is also the practical implementation of the affine structure.
In practice, one takes a couple of rulers and moves a ruler always remaining in contact with the other, kept at rest. And next do the same with the latter ruler, initially at rest, keeping at rest the former.
This way to explore the space shows that the geometry seen by this class of rulers is, with an extrarodinary precison at human scales at least, the one of Euclid: $(\mathbb{A}^3, -, V^3, \cdot)$.
Another crucial fact is that one may next invent other sort of "rulers" using other part of physics like optiks -- for instance exploring  smaller or larger scales -- and, always, the measurements agree with those of the initial rigid rulers -- at intermediate scales where both procedures can be exploited.
All that means that our physics embodies a unique geometry and this is  Euclid's one.
As is known, at large scales things change dramatically. But that's another story.
