Galilean Coordinate Space

up:: Galilean Space

A galilean coordinate space is the Canonical Affine Structure of a Vector Space $\mathbb{R}\times {\mathbb{R}}^3$ from $({\mathbb{R}}^4,+,\cdot )$, where we consider $({\mathbb{R}}^3,⟨\cdot ,\cdot ⟩)$ an Euclidean Vector Space.

Note that this space is trivially a Galilean Space, since

1. Every vector space induces an affine space¹
2. One can define a time functional as merely the distance in the first $\mathbb{R}$ axis². Formally,

$$\begin{array}{rl}t:(\mathbb{R}\times {\mathbb{R}}^3,+)& \to \mathbb{R}\\ (\Delta t,\Delta \vec{x})& \mapsto \Delta t\end{array}$$

3. The distance between simultaneous events can be created from the Vector Space Norm induced by the Inner Product from the Euclidean Structure

Possible transformations of this space

• Galilean boosts preserve the Galilean Structure

Translations preserve galilean structure

Since Translations in affine spaces are affine automorphisms, every translation $T(t,\vec{x})=(t+\tau ,\vec{x}+\vec{v})$.

Rotations of the coordinate axes preserve galilean structure

Rotation of the coordinate axes: $\tilde{R}(t,\vec{x})=(t,R\vec{x})$, where $R\in O(3)$ is an Orthogonal Transformation

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References

• ARNOL’D, Vladimir Igorevich. Mathematical methods of classical mechanics. Springer Science & Business Media, 2013.
• Notes on Mathematical Physics for Mathematicians - Daniel Tausk

Footnotes

1. Since Fixing a point in an affine space induces a vector space. ↩

2. Thinking about $(\mathbb{R}\times {\mathbb{R}}^3,+,\cdot )$ as the underlying vector space to the set $\mathbb{R}\times {\mathbb{R}}^3$. ↩