Apologies as I am quite unfamiliar with rotational physics. But I hope that some physicists here may be able to give me a satisfying and intuitive explanation.
Background/Motivation: I am from a pure mathematics background, thinking about the curl for vector fields in 3D there's a geometric interpretation of curl of a vector field $F: \mathbb R^3 \to \mathbb R^3$ in this video (near the very end) https://www.khanacademy.org/math/multivariable-calculus/multivariable-derivatives/curl-grant-videos/v/3d-curl-intuition-part-2 : take a small sphere with center fixed at some point, and let vector field $F$ "act on" the sphere [I do not know rigorously how to define this]. This will cause the sphere to rotate around some axis with some rate, which can be described by a vector (axis of right-hand-rule rotation + magnitude of rotation). This vector (which I'll call the "vector of rotation", denoted $\vec v_F$) is precisely the curl of $F$ (maybe up to a factor of $2$).
This geometric interpretation is also described in writing in this article https://mathinsight.org/curl_idea
((( For curl, since it is a local phenomenon and independent of adding a constant to the v.f. $F$, it suffices to understand this interpretation for $F: \mathbb R^3 \to \mathbb R^3$ a linear transformation. (Since zooming in locally at say $0$ where we normalize $F(0)=0$, one can not tell the difference between $F:\mathbb R^3 \to \mathbb R^3$ and $dF: \mathbb R^3 \to \mathbb R^3$ at $0$) )))
Question:
The question I have for physicists is about linearity of the "rotation vector" map $\rho: F \mapsto \vec v_F$, i.e. if I "act on" the small sphere with the sum of 2 velocity vector fields $F+G$, why is it the case that the resulting vector of rotation exactly equal to $\vec v_F + \vec v_G$?
(From linearity, deriving the differential formula for curl is not difficult https://mathinsight.org/curl_components.)
Looking at explanations online, people talk about angular momentum, torque, etc. From a pure mathematics background, I am not so happy with these formulas. I am wondering if there is a symmetry based approach, that allows to derive the formulas for angular momentum, torque, etc. from a very small set of minimal natural physical assumptions (like invariance/equivariance under symmetry), or enough so that one can prove linearity of the rotation vector map $\rho$.
For instance I know maybe Noether's theorem gives us a "mathematically satisfying" definition of angular momentum [assuming some Lagrangian or Hamiltonian formalism, which I am hesitant to assume...I would prefer as simple an explanation as possible], but then why are we sure that that mathematical formula, is precisely the axis and magnitude by which the small sphere will actually physically rotate? [This is largely a rhetorical question; but the point is just that if one chooses to explain things in whatever way, an important piece is the bridge between whatever fancy mathematics you use and the actual physical observables]
My hope is some explanation similar to the simple symmetry-based "2 clay ball" argument for kinetic energy's quadratic dependence on velocity given in this classic answer.
EDIT: my friend H gave the following very beautiful argument. I am not sure if there is more to say, but certainly it is quite something. Let us consider "forces"/"velocity actions" on 2 points of the sphere. I think it is acceptable to say that the only contributions of the "forces" on the sphere are the projections of the forces to the tangent planes to the sphere, i.e. only tangential components matter. First, look at each "force" $\vec F_a$ at the point $a$ in isolation. Surely, this will cause the sphere to rotate, on the axis perpendicular to both [the vector from center of sphere to point $a$] and [$\vec F_a$ tangent to the sphere]. While it rotates, $a$ will sweep out a great circle $C_a$.
H's argument is as follows: by symmetry, it shouldn't matter where the point $a$ is on the great circle $C_a$. Similarly, considering $b$ and $\vec F_b$ in isolation, it should also not matter where $\vec F_b$ applies on the great circle $C_b$. So, we can simply parallel translate both $\vec F_a, \vec F_b$ along their respective great circle, until they lie at the same point (any 2 great circles on the sphere intersect!!!). And then, if we accept additivity of "forces" at a single point, then we should also accept additivity of "torque" at 2 different points.
Then, for $n$ points, simply perform this trick moving (tangential) forces along their great circles, and you can turn 2 forces at 2 points into 1 summed force at 1 point. So you can just keep combining pairs of points, until you get all the way down to 1 point.
[Of course, this argument requires one to accept that because scenarios $\Sigma_a, \Sigma_a'$ are "physically identical" looking at each $a$ (or $b$) in isolation, then considering $a,b$ together, then $\Sigma_a, \Sigma_b$ put together is "physically identical" to $\Sigma_a', \Sigma_b'$ put together. But maybe this is a foundational physical assumption?]
