Moving objects change their position over time with respect to a reference frame. By reference frame, we mean a set of directions(axes) and a reference point(origin).

We should select a frame of reference in which Newton’s first law is valid i.e., if no force acts on an object, the object continues its state of uniform motion. Such a reference frame is called inertial frame of reference. The ground is most convenient choice for most of everyday problem analysis.

cartesian coordinates

Usually, the set of axes are perpendicular to each other and coincide at the origin. We call the origin O and the set of axes x, y and z. Once you have the set of axes and the origin in place, you can define the position of anything in space by mentioning its distance from the three axes and the origin. We call them coordinates. This system of is called Cartesian system of coordinates and it is the most popular type of coordinates. Other systems include spherical polar coordinates, cylindrical polar coordinates etc.

The concept of inertial frame of reference is basically an idealization. In Galilean formulation, a fundamental assumption stands that

A frame of reference can always be chosen in which space is homogenous and isotropic and time is homogenous as well; we call it the inertial frame. There are infinitely many inertial frames moving relatively with constant velocity or oriented differently with respect to each other. All of these have same laws of mechanics and same properties of space and time. Therefore, there is no absolute frame of reference.

– Mechanics: Landau and Lifshitz

Any inertial frame is never unique. For example, a frame A moving with constant velocity v with respect to an inertial frame S, is also an inertial frame of reference.

At least 10 linearly independent transformations (getting a new inertial frame S’ by performing operation on S) S-> S’ hold:

  1. 3 rotations: r’ = O r; O is a 3×3 orthogonal matrix
  2. 3 translations: r’ =c; c is a constant vector
  3. 3 boosts: r’ = + ut; u is constant velocity vector
  4. 1 time translation: t’ = t+c; c = some real number

(r: position vector in S, r’: position vector in S’)

If motion is uniform in S, it will also be uniform in S’. These transformations make up the Galilean Group under which Newton’s laws are invariant.

Reference frames
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