I was reading Arnold's Mathematical Methods in Classical Mechanics and it's not clear to me what exactly he's trying to achieve by introducing an affine space to model the set of all "positions in the universe". It's certainly a more intuitive model of the world that doesn't assume more structure than what's required but right after introducing the affine structure, he moves on to define "motion" as a smooth map $\textbf{x}:I \longrightarrow \mathbb{R}^3 $ from some interval $I$ to the vector space $\mathbb{R}^3 $. The velocity and acceleration are defined as derivatives of this map. After doing so I don't think he refers to the Affine structure anywhere else in the book and he only talks about the map $\textbf{x}$.

Questions -

1) So what exactly is the motion map $\textbf{x}$ supposed to be? Is it to be interpreted as some sort of displacement vector that evolves over time and gives us the net displacement of a particular particle? Moreover is this an element of the vector space that acts on the affine space to produce translations?

2) What exactly is the structural difference brought about by having an affine space of positions rather than a vector space? Is it just so that we assume less structure or is there some genuine motivation for introducing it?


1 Answer 1


Arnold's aim is to model space and spacetime in a coordinate-free way. To achieve this, he identifies locations and events as points in abstract affine spaces $A^n$ ($n=3,4$ respectively).

The problem is, when you remove coordinates it gets very hard to define many important dynamical concepts and quantities (e.g. force and acceleration) without becoming excessively abstract.

To get around this, you set up a bijection between the abstract affine space $A^n$ and a concrete (coordinatized) affine space $\mathbb{R}^n$. You can think of the latter as a numerical grid placed over the top of the former. You can now define your dynamical constructs in terms of coordinates (taking advantage of calculus, for example), and define their abstract counter-parts indirectly.

To use your example, a motion (of a single particle) is defined by Arnold as a map $\textbf{x}: I \longrightarrow \mathbb{R}^3$, which is obviously a definition in coordinate space. However, if you read on, you'll see that he describes a world line as a curve in affine space which appears as a motion in coordinate space.

A world line is thereby defined in a coordinate-free way by using the concept of a motion as scaffolding. This gives you an idea of how abstract coordinate-free definitions are often obtained by 'pulling back' definitions given in canonical coordinate spaces.

It's worth getting your head around this, because it is precisely how things proceed in the subject of Differential Geometry, where coordinate-free manifolds (e.g. representing space and spacetime) are given all sorts of elaborate structures by working in a coordinate image and then pulling back the result to the manifold (and vice-versa).

Whilst you are not wrong to think of a motion as a continuous succession of displacement vectors, the interpretation I outline here will prepare you much more adequately for the study of differential geometry, which is at the heart and soul of virtually all of physics.


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