February 15, 2022

Inertial reference frame: A frame in which a particle can only accelerate via its interaction with some other object. That is, the cause of acceleration is due to physical interaction, in other words, in absence of any acceleration causing agents in the frame, the body follows the path defined by its inertia.

All the inertial frames are equivalent to each other in all mechanical respects, that is there is no experiment that one can do to tell if they are moving with uniform velocity or not. This is the essence of the Galilean principle of relativity. Consider an inertial frame K’ (with axes x’, y’, z’) moving WRT to inertial frame K (with axes x, y, z that are parallel to axes x’, y’, z’ respectively) such that the velocity of K’ is in the x-axis direction. Taking some Point A, whose position vector in K and K’ is respectively $\vec{r}$ and $\vec{r}'$, then taking $\vec{r}$ = $\vec{r}'$ at $t$ = 0, we get

$$ \vec{r}'=\vec{r}-\vec{V}t \tag{2.1} $$

where $\vec{V}$ is the velocity of K’ WRT K. And also

$$ t'=t\tag{2.2} $$

That is time rate is independent of the frame of reference.

Equations (2.1) and (2.2) are Galilean transformations used to locate the position of an object from one inertial reference frame to the other.

Differentiating equation (2.1) WRT time we get

$$ \vec{v}'=\vec{v}-\vec{V}\tag{2.3} $$

Differentiating equation (2.3) WRT time and recalling that $\dot{\vec{V}}$ = 0, since K’ is inertial, we get

$$ \vec{a}'=\vec{a}\tag{2.4} $$

That is, if a particle accelerates with some acceleration $\vec{A}$, in one inertial frame, then it accelerates with the same acceleration in all inertial frames.

In other words, an acceleration caused by a physical interaction is independent of the choice of inertial frame.

Fundamental laws of Newtonian mechanics :