Physical space

Physics
Mechanics
Introduction to affine spaces

Introduction

It’s quite common to think of physical space as \(\mathbb{R}^3\), but this perspective can be misleading, both from a mathematical and a physical standpoint. Instead, the three-dimensional space we encounter in Classical Physics (and even the four-dimensional spacetime in Special Relativity) is more accurately captured by the concept of affine spaces.

Definition

To understand this better, let’s dive into what a real affine space is. A real affine space of finite dimension \(n\), denoted by \(\mathbb{A}^n\), is a collection of elements known as points, but with some additional structure:

  • There’s an associated \(n\)-dimensional vector space \(V\), which we call the space of displacements or the space of free vectors.
  • A mapping \(\mathbb{A}^n \times \mathbb{A}^n \ni (P, Q) \mapsto P - Q \in V\) that respects certain conditions:
    • For every point \(Q\in \mathbb{A}^n\) and every vector \(\mathbf{v} \in V\), there’s a unique point \(P \in \mathbb{A}^n\) such that \(P - Q =\mathbf{v}\).
    • For any three points \(P, Q, R \in \mathbb{A}^n\), the equation \(P - Q + Q - R = P - R\) always holds.

Example

A trivial example is the real affine space \(\mathbb{A}^n = \mathbb{R}^n\) with the displacement space being the vector space \(\mathbb{R}^n\). A more interesting example is the system \(\begin{cases}x + y + z = 2 \\2x + y - z = 1\end{cases}\)

a = matrix([[1,1,1], [2,1,-1]])
b = vector([2, 1])

The solution space of the system is an affine space \(\mathbb{A}^3\):

A = AffineSpace(3, RR)
A
\(\displaystyle \newcommand{\Bold}[1]{\mathbf{#1}}\mathbf{A}_{\Bold{R}}^{3}\)

Additionally, the the solution space of the homogeneous system is the space of displacements \(V\):

V = VectorSpace(RR, 3).subspace(a.right_kernel().basis())
V
\(\displaystyle \newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{RowSpan}_{\Bold{R}}\left(\begin{array}{rrr} 1.00000000000000 & -1.50000000000000 & 0.500000000000000 \end{array}\right)\)

We can get a particular solution \(Q \in \mathbb{A}^3\) and verify that it is indeed a solution:

Q = A.point(tuple(a.solve_right(b)))
assert a * vector(Q) == b
Q
\(\displaystyle \left(-1.00000000000000, 3.00000000000000, 0.000000000000000\right)\)

The general solution of the system is given by the displacement of \(Q\). For example another solution \(P \in \mathbb{A}^3\) is the following:

v, *_ = V.basis()
P = A.translation(-v)(Q)
assert a * vector(P) == b
P
\(\displaystyle \left(0.000000000000000, 1.50000000000000, 0.500000000000000\right)\)