Reference Frames

In classical mechanics, we assumed that it is possible to assign a 3-dimensional real vector to each point in space and a real number to each moment in time. However, there seem to be infinitely many ways to do this. For example, one might assign the real vector $0$ to the top-right corner of the screen of my monitor, or to the point of the North Pole or even to the center of the Sun. Similarly, one might choose to assign a time of $0$ to the moment of their birth. In this case, moments before their birth would be assigned negative numbers and moments after would be assigned positive numbers. But one can just as well choose to assign $0$ to the moment the first star was formed or to the moment when the star Betelgeuse will explode into a supernova.

The point is that there are infinitely many ways to perform such assignments and so we have the freedom to choose any one of them. In most cases, this choice is irrelevant but, sometimes, it can be a powerful way of simplifying a given problem.

Definition: Reference Frame

A reference frame is a choice of a way to assign a 3-dimensional real vector to each point in space and a real number to each moment in time.

Definition: Origin

The origin of a reference frame is the point in space which is assigned the zero vector $0$.

Note: Temporal Origin

Sometimes, temporal origin is used to refer to the moment which gets assigned a time of $0$.

By far the most common way of choosing a reference frame is to pick a point for the origin and then choose three perpendicular axis along which the components of the assigned vectors increase / decrease.

Important: Reference Frame Attachment

It is very common to “attach” a reference frame to a specific object. This just means that, as time moves forward, we update our reference frame so that its origin is always at the object’s location.