Every Galilean transformation can be written as the composition of rotation, translation, and uniform motion

Question

Having heard many good things about Arnold's Mathematical Methods of Classical Mechanics, I picked it up and started going through it. While I think I understand all of the definitions he makes, the problems (at least at the beginning) are very mathematical and my proof skills aren't all that great. Can someone show me how to prove this one for instance?

Problem: Show that every Galilean transformation of the space $\Bbb R \times \Bbb R^3$ can be written in a unique way as the composition of a rotation, a translation, and a uniform motion (thus the dimension of the Galilean group is equal to $3+4+3=10$).

Here are some of the relevant definitions:

Definition: Galilean space: An affine space $A^4$ equipped with a Galilean space-time structure.

Definition: Galilean structure: The following properties:

1. parallel displacements of $A^4$ constitute a vector space $\Bbb R^4$.
2. The linear mapping $t:\Bbb R^4 \to \Bbb R$ from the vector space of parallel displacements of $A^4$ to the real axis is given by $t(b-a)$ for $a,b \in A^4$ and is called the time interval. If $t(b-a)=0$ then the events $a$ and $b$ are called simultaneous.
3. The distance between simultaneous events $$d(a,b) = \|a-b\| = \sqrt{(a-b,a-b)} \tag{a,b \in A^3}$$ is given by a scalar product on the space $\Bbb R^3$. This distance makes every space of simultaneous events into a $3$d Euclidean space.

Definition: Galilean Group: The group of all transformations of a Galilean space which preserve its structure (preserves intervals of time and distance between simultaneous events).

Defintion: Galilean transformation: An element of the Galilean group.

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Two addition questions for others who have read Arnold's book: $1)$ should I try to learn some more abstract math (like maybe abstract algebra) before attempting to read this book? I have already taken all of the math requirements I have as an undergrad -- multivariable calculus, ODEs, PDEs, linear algebra, and a math methods course. $2)$ Is there a solutions manual available for this book?

Answer 1

Let $\phi:\mathbb{R}^3\times\mathbb{R}\rightarrow\mathbb{R}^3\times\mathbb{R}$ be a Galilean transformation. Let $R\in\mathrm{O}(3)$, $\tau\in\mathbb{R}$ and $\mathbf{v},\mathbf{y}\in\mathbb{R}^3$. The goal of this exercise is to show that $$\phi: \begin{pmatrix} \mathbf{x} \\ t \end{pmatrix}\mapsto \begin{pmatrix} R & \mathbf{v} \\ 0 & 1 \end{pmatrix}\begin{pmatrix} \mathbf{x} \\ t \end{pmatrix}+\begin{pmatrix} \mathbf{y} \\ \tau \end{pmatrix} \tag1 $$ and that this is unique. We first note that the rotation group in $\mathbb{R}^3$ is three dimensional. We then have the boost and the spatial translation, which gives $2\cdot 3=6$. Plus the time translation, we find that the Galilean group is $10$-dimensional.

The most general $\mathbb{R}$-affine map is $$\phi(\mathbf{x},t)=A(\mathbf{x},t)+(\mathbf{y},\tau)$$ Let us write this out in "components": $$A(\mathbf{x},t)=(A_{11}\mathbf{x}+A_{12}t,A_{21}\mathbf{x}+A_{22}t)$$ In order that lengths be preserved, we require $A_{11}$ be an orthogonal matrix. Differences between times of events should also be preserved. Suppose we have two events $x_1$ and $x_2$. This condition means that $$A_{22}(t_2-t_1)+A_{21}(\mathbf{x}_2-\mathbf{x}_1)=t_2-t_1$$ For this to hold at all $\mathbf{x}_1$ and $\mathbf{x}_2$, we require $A_{21}=0$ which implies $A_{22}=1$. We identify the remaining component of $A$ as the linear map $\mathbb{R}\rightarrow\mathbb{R}^3$, which is the velocity.

We have thus shown that the most general Galilean transformation is of the form $(1)$.

For more information and more detailed derivations, see this website and these lecture notes.