Comments about "Galilean transformation" in Wikipedia

This document contains comments about the article Galilean transformation in Wikipedia
In the last paragraph I explain my own opinion.

Contents

Reflection

Introduction

The article starts with the following sentence.

1. Translation

Though the transformations are named for Galileo, it is absolute time and space as conceived by Isaac Newton that provides their domain of definition.
Isaac Newton only used the concepts time and space.
In essence, the Galilean transformations embody the intuitive notion of addition and subtraction of velocities as vectors.
In the physical reality only positions and time make sense. A velocity is a mathematical (calculated) concept. This has nothing to do with intuition.
The notation below describes the relationship under the Galilean transformation between the coordinates (x, y, z, t) and (x', y', z', t') of a single arbitrary event, as measured in two coordinate systems S and S', in uniform relative motion (velocity v) in their common x and x' directions, with their spatial origins coinciding at time t = t' = 0:
See Reflection 1 - An exercise with Coordinates

2 Galilean transformations

3 Galilean group

4 Origin in group contraction

5 Central extension of the Galilean group

6. See also

Following is a list with "Comments in Wikipedia" about related subjects

Reflection 1 - An exercise with Coordinates

A coordinate system is used to describe the position of objects at a certain moment t. To do that the coordinate system consists of three axis x,y,z perpendicular towards each other. The center is called the origin. The origin is normally a certain reference point or reference object. The time of the origin is identified as t=0. In general the position of an object is described as (x,y,z,t). That means at time t the object is at position (x,y,z)
Using a coordinate system it is possible to describe the behaviour of certain object.
Such a description starts with observations.

Example 1:
In this particular example only the x coordinate is used.
The first observation is: at t1=0 object A is at position x1 = 0.
The second observation is: at t1= 10 object A is at position x2 = 20.
Using the equation vx = (x2 - x1)/(t2-t1), shows that vx = (20 - 0)/(10 - 0) = 20/ 10 = 2 This means that object A has a speed 2 in the x direction.

Example 2:
In this particular case also only the x coordinate is used.
The first observation is: at t1=0 object B is at position x1 = 100.
The second observation is: at t1= 10 object B is at position x2 = 80.
Using the equation vx = (x2 - x1)/(t2-t1), shows that vx = (80 - 100)/(10 - 0) = -20/ 10 = -2 This means that object B has a speed -2 in the x direction.

Example 3:
This is a combination of example 1 and example 2. At t = 0 object A is at position x1 = 0 with a speed vx = 2 and object B is at position x1 = 100 with a speed vx = -2.
That means the two objects approach each other; there will be a collision at position xc
For object A xc = x1 + vx*t = 0 + 2*t.
For object B xc = x1 + vx*t = 100 - 2*t
This gives xc = 0 + 2*t = 100 - 2*t or 4*t = 100 or t = 25
That means there is collision at t = 25 and the distance xc = 50

What is the importance of this whole exercise?
In this whole exercise there are two moving objects involved and based on observations the speed of both objects are calculated. Only one coordinate system is used. And what is also important no direct addition of velocities is used. This is important because addition of velocities is tricky.
From the point of view of an observer in object A, object B is approaching at a speed of vx = 4. From the point of view of observer B the same is happening: object A is approaching at a speed of vx = 4. This is mathematical correct, but physical wrong. In neither experiment a physical speed of v = 4 is involved. Only a speed of vx =2 (towards the right) in the x direction and a speed of vx=-2 (towards the left) in the x direction That is the most important physical lesson in the exercise.

Reflection 2 - Transformations

The reason behind Reflection 1 - An exercise with Coordinates is to chalenge the importance of coordinate transformations. A coordinate system is a mathematical construction which are linked to a certain reference point or object, but they have no physical speed.

Reflection 3

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Created: 17 Augustus 2019

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