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 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.