Galilean Transformations

We will be using the math developed in our blog post Galilean Story.

Our goal will be to start with $\overrightarrow{S} = t \overrightarrow{e}_t + x \overrightarrow{e}_x$ and finish with $\overrightarrow{S} = \tilde{t} \overrightarrow{e'_t} + \tilde{x} \overrightarrow{e'_x}$ .

The equations are shown below.

$\overrightarrow{e'}_t = \overrightarrow{e_t} + v \overrightarrow{e_x}$

$\overrightarrow{e'}_x = \overrightarrow{e_x}$

$\overrightarrow{e}_t = \overrightarrow{e'_t} - v \overrightarrow{e'_x}$

$\overrightarrow{e}_x = \overrightarrow{e'_x}$

$\tilde{t} = t$

$\tilde{x} = -vt + x$

We are going to take the equations above and start with…

$\overrightarrow{S} = t \overrightarrow{e}_t + x \overrightarrow{e}_x$

…and then make substitutions to slowly change the non-primed vectors and components to primed vectors and components, one equation at a time. When we are finished we will have $\overrightarrow{S}$ as it is expressed in the primed coordinate system.

When we look at the last equation…

$\overrightarrow{S} = \tilde{t} \overrightarrow{e'_t} - v \tilde{t} \overrightarrow{e'_x} + \tilde{x} \overrightarrow{e'_x} + vt \overrightarrow{e'_x}$