Show that all galilean spaces are isomorphic to each other \({ }^{6}\) and, in particular, isomorphic to the coordinate space \(\mathbb{R} \times \mathbb{R}^{3}\). Let \(M\) be a set. A one-to-one correspondence \(\varphi_{1}: M \rightarrow \mathbb{R} \times \mathbb{R}^{3}\) is called a galilean coordinate system on the set \(M\). A coordinate system \(\varphi_{2}\) moves uniformly with respect to \(\varphi_{1}\) if \(\varphi_{1} \varphi_{2}^{-1}: \mathbb{R} \times \mathbb{R}^{3} \rightarrow \mathbb{R} \times \mathbb{R}^{3}\) is a galilean transformation. The galilean coordinate systems \(\varphi_{1}\) and \(\varphi_{2}\) give \(M\) the same galilean structure.

In classical mechanics, the Galilean coordinate system is a tool we use to describe the position and motion of objects. Galilean coordinate systems are made up of a time axis and three spatial dimensions, which we commonly notate as \( \mathbb{R} \times \mathbb{R}^{3} \). This representation allows us to describe the position of any point or object at a given time.

When we talk about a one-to-one correspondence with a set \( M \), what we mean is that there is a unique mapping for every point in \( M \), with no repeats or omissions. This mapping connects each point in \( M \) to a specific point in time-space, \( \mathbb{R} \times \mathbb{R}^{3} \)—the realm of everyday distances and durations. Thus, a coordinate system is simply a way to link the abstract points in \( M \) to concrete positions and moments we can measure and comprehend.

A Galilean transformation is the shift from one Galilean coordinate system to another while preserving the structure of space and time laid out by Newton's laws of motion. Essentially, it's the mathematical framework that we use to switch perspectives between different observers moving at constant velocities relative to each other, without altering the physics of the scenario.

For two coordinate systems, \( \varphi_{1} \) and \( \varphi_{2} \) moving uniformly with respect to one another, the transformation \( [\varphi_{1} \varphi_{2}^{-1}]: \mathbb{R} \times \mathbb{R}^{3} \rightarrow \mathbb{R} \times \mathbb{R}^{3} \) is our Galilean transformation. It ensures that the fundamental principles, like the relativity of motion and the independence of space and time, remain intact despite the change in the observer's motion.

Mathematical isomorphism is a concept that enables us to say two mathematical structures are, for all intents and purposes, the same. It's a bijective correspondence—meaning a pairing where each element of one set is matched with a unique element of another set, and vice versa—that plays nicely with the operations and relations of the sets involved.

So, when we talk about Galilean spaces being 'isomorphic', we're saying they can be perfectly mapped onto each other via a bijective function, preserving the structure. It means no matter which Galilean space (a kind of mathematical model of reality) you're working with, you can expect it to behave in the same way as any other Galilean space, or even the standard coordinate space \( \mathbb{R} \times \mathbb{R}^{3} \). This concept is what underpins the simplicity and elegance of classical mechanics, allowing physicists and mathematicians to easily translate problems into a standard form where they're more manageable and intuitive to solve.