Galilean transformation

Question:
Physics laws cannot be altered by the observer's frames of reference. The transformation from a lab reference  frame to a moving reference frame is known as Galilean transformation. Show that Newton's laws are invariant under Galilean transformation.

Answer:
Galilean transformation subtracts the amount of displacement regarding velocity of the moving frame of reference to make the equivalent observation from the lab frame. Namely,
\[
x' = x - vt, \quad t' = t
\]
We assume that the time elapses equally for both frames; and the relative velocity is constant. Thus, the velocity becomes
\[
v' = \frac{dx'}{dt'}=\frac{d}{dt}(x-vt)=\frac{dx}{dt}-v
\]
The acceleration becomes
\[
a' = \frac{d^2x'}{dt'^2}=\frac{dv'}{dt'}=\frac{d}{dt}\left(\frac{dx}{dt}-v \right)=\frac{d^2x}{dt^2}=a
\]
Again, note that $v$ is a constant. Therefore, the Newton's equation of motion (laws) must be invariant under Galilean transformation.
\[
F'=m\frac{d^2x'}{dt'^2}=ma=F
\]
In other words, the physics law is the same from any observers.

Powered by Hirophysics.com