Question

In Arnold's Mathematical Methods of Classical Mechanics, we define the physical space time as a four dimensional affine space with associated Galilean structure. I understand this part.

Now what I'm not clear after reading the next section in the book is:

What's the definition of a frame of reference, i.e. the formal definition of the mapping from the physical spacetime to R4. I guess the mapping needs to be bijective and preserving the Galilean structure.

What's the definition of an inertial frame? Is it an additional axiom that states there's a special frame mapping? Or does the concept of inertial frame already emerge from the Galilean spacetime definition?

Answer 1

A frame of reference is a coordinate system that defines the positions and velocities of objects in a physical space-time. An inertial frame of reference is a special type of frame in which Newton's First Law of Motion holds true.

Step-by-step explanation:

A frame of reference is a coordinate system that defines the positions and velocities of objects in a physical space-time. It is a way to describe and compare the motion of objects with respect to a particular observer or reference point. In the context of classical mechanics, a frame of reference is typically defined by a mapping from the physical space-time to a four-dimensional affine space, such as 4.

An inertial frame of reference is a special type of frame in which Newton's First Law of Motion holds true. In other words, objects at rest remain at rest, and objects in motion continue to move with constant velocity in a straight line, unless acted upon by an external force. It is not an additional axiom, but rather a property that emerges from the definition of the Galilean structure of the physical space-time.