Galilean Transformations#

It is quite common for some physical problems to be easier to solve in one reference frame than in another or for us to want to find kinematical quantities in different reference frames. To achieve this, we need a way to convert between kinematical quantities in one reference frame to the kinematical quantities in another reference frame.

In classical mechanics, this conversion is done using Galilean transformations.

Axiom: Galilean Transformations

Let $\mathcal{R}_{\alpha}$ and $\mathcal{R}_{\beta}$ be reference frames with origins $\mathcal{O}_{\alpha}$ and $\mathcal{O}_{\beta}$ and temporal origins $T_{\alpha}$ and $T_{\beta}$, respectively. We assume the following:

If a moment in time is ${}_{\mathcal{R}_{\alpha}}t$ in $\mathcal{R}_{\alpha}$, then in $\mathcal{R}_{\beta}$ it is
$$
{}{\mathcal{R}}}t = {{\mathcal{R}}}t + T_{\alpha} - T_{\beta
$$
If a point particle $p$ has position ${}_{\mathcal{R}_{\alpha}} \boldsymbol{r}_p$ at time ${}_{\mathcal{R}_{\alpha}}t$ with respect to $\mathcal{R}_{\alpha}$ and if the position of $\mathcal{O}_{\beta}$ at time ${}_{\mathcal{R}_{\alpha}}t$ with respect to $\mathcal{R}_{\alpha}$ is ${}_{\mathcal{R}_{\alpha}} \boldsymbol{r}_{\mathcal{O}_{\beta}}$, then the position of $p$ at time ${}_{\mathcal{R}_{\beta}}t$ with respect to $\mathcal{R}_{\beta}$ is

\[ {}_\mathcal{\mathcal{R}_{\beta}} \boldsymbol{r}_p = {}_\mathcal{\mathcal{R}_{\alpha}} \boldsymbol{r}_p - {}_{\mathcal{R}_{\alpha}} \boldsymbol{r}_{\mathcal{O}_{\beta}} \]

If the velocity of $\mathcal{O}_{\beta}$ at time ${}_{\mathcal{R}_{\alpha}}t$ with respect to $\mathcal{R}_{\alpha}$ is ${}_{\mathcal{R}_{\alpha}}\boldsymbol{v}_{\mathcal{O}_{\beta}}$ and the velocity of $\mathcal{O}_{\alpha}$ at time ${}_{\mathcal{R}_{\beta}}t$ with respect to $\mathcal{R}_{\beta}$ is ${}_{\mathcal{R}_{\beta}}\boldsymbol{v}_{\mathcal{O}_{\alpha}}$, then

\[ {}_{\mathcal{R}_{\beta}}\boldsymbol{v}_{\mathcal{O}_{\alpha}} = -{}_{\mathcal{R}_{\alpha}}\boldsymbol{v}_{\mathcal{O}_{\beta}} \]

The above equations are called Galilean transformations.

From the above equations we can derive two other equations regarding velocity and acceleration.

Theorem: Velocity and Acceleration Addition

Let $\mathcal{R}_{\alpha}$ and $\mathcal{R}_{\beta}$ be reference frames with origins $\mathcal{O}_{\alpha}$ and $\mathcal{O}_{\beta}$ and let $p$ be a point particle. Suppose we have a moment in time which is ${}_{\mathcal{R}_{\alpha}}t$ in $\mathcal{R}_{\alpha}$ and ${}_{\mathcal{R}_{\beta}}t$ in $\mathcal{R}_{\beta}$.

If the velocity of $p$ with respect to $\mathcal{R}_{\alpha}$ at ${}_{\mathcal{R}_{\alpha}}t$ is ${}_{\mathcal{R}_{\alpha}}\boldsymbol{v}_{p}$ and the the velocity of $\mathcal{O}_{\beta}$ with respect to $\mathcal{R}_{\alpha}$ at ${}_{\mathcal{R}_{\alpha}}t$ is ${}_{\mathcal{R}_{\alpha}}\boldsymbol{v}_{\mathcal{O}_{\beta}}$, then the velocity of $p$ with respect to $\mathcal{R}_{\beta}$ at ${}_{\mathcal{R}_{\beta}}t$ is

\[ {}_{\mathcal{R}_{\beta}}\boldsymbol{v}_{p} = {}_{\mathcal{R}_{\alpha}}\boldsymbol{v}_{p} - {}_{\mathcal{R}_{\alpha}}\boldsymbol{v}_{\mathcal{O}_{\beta}} \]

If the acceleration of $p$ with respect to $\mathcal{R}_{\alpha}$ at ${}_{\mathcal{R}_{\alpha}}t$ is ${}_{\mathcal{R}_{\alpha}}\boldsymbol{a}_{p}$ and the the acceleration of $\mathcal{O}_{\beta}$ with respect to $\mathcal{R}_{\alpha}$ at ${}_{\mathcal{R}_{\alpha}}t$ is ${}_{\mathcal{R}_{\alpha}}\boldsymbol{a}_{\mathcal{O}_{\beta}}$, then the acceleration of $p$ with respect to $\mathcal{R}_{\beta}$ at ${}_{\mathcal{R}_{\beta}}t$ is

\[ {}_{\mathcal{R}_{\beta}}\boldsymbol{a}_{p} = {}_{\mathcal{R}_{\alpha}}\boldsymbol{a}_{p} - {}_{\mathcal{R}_{\alpha}}\boldsymbol{a}_{\mathcal{O}_{\beta}} \]

Proof

TODO

Galilean transformations are true as long as we are not dealing with very high speeds, way below $300\,000\,000 \frac{\mathrm{m}}{\mathrm{s}}$. Reality, however, is more complicated as you approach these speeds because, for some reason, $300\,000\,000 \frac{\mathrm{m}}{\mathrm{s}}$ is a universal speed limit. In these situations, Galilean transformations break down and we need other transformations in order to match empirical observation.