Frames Of Reference and Co-ordinate Systems

Concept of frames of reference is one of the most fundamental ideas in classical mechanics which is of pivotal importance in an attempt to understand the working mechanism behind classical mechanics. Classical Mechanics, at intermediate level, consists of a small set of concepts like inertial and non-inertial frames of references, definitions of velocity and acceleration, Newton’s laws of motion, conservation of momentum and mechanical energy and the concept of centre of mass/gravity. The rest of the part is about applying these concepts using mathematics to solve or analyze classical systems. Therefore it is incumbent that we be conversant with these fundamental concepts.

Frames of Reference and Co-ordinate Systems

In classical mechanics, one’s aim is to be able to describe and predict the future course of motion of a system. The motion of a body can only be observed/described relative to something - other bodies, observers, or a set of space-time coordinates. These are called frames of reference. Our observations of the motion of a system will depend upon how we choose our frame of reference to make observations.

Thus, a frame of reference is related to the state of motion of the observer. On the other hand, a co-ordinate system is a scheme to represent the observations in mathematical terms. A co-ordinate system is attached to a frame of reference (an observer) and serves to make measurements. Measurements can be made in terms of distances along rectilinear (straight) directions (which we treat as axes), or in terms of angles about these directions, or when can choose the axes to be curvilinear instead of straight lines. There are different kinds of co-ordinate systems corresponding to different ways used to make measurements. A co-ordinate system is chosen in such a way so as to make a problem mathematically simpler.

For example, consider an observer standing still at the centre of a ground observing a bike going in a circle of radius R with a constant velocity with the observer at the centre of the circle. Now in order to describe this circular motion the observer will have to first represent the position of the bike in some mathematical form, i.e., he will have to assign certain set of co-ordinates to the bike to locate its position anywhere on the ground. This can be done in various ways.

One possibility is that he can adopt rectangular co-ordinate system where he can consider himself to be at the origin, define x- and y- axes and then locate the bike at a given instant by listing its co-ordinates (x, y) at that instant. The trajectory of the bike (or its equation of motion) will be

Differentiating the above equation with respect to time gives relation between the x and y components of the velocity :

 

Instead, the observer can also choose to locate the bike in terms of the radius of the circle R and the angle θ that the bike makes with the x- (or y- ) axis. Here the observer will list the co-ordinates as (R, θ). This kind of co-ordinate system is called Polar Co-ordinate System. Since the bike is going in a circle of a constant radius the change in the radius throughout the motion is 0 i.e. the R co-ordinate for all the positions of the bike remains the same. Now we need to worry only about the angle θ of the bike. Since the bike is moving with a constant velocity, its angular velocity (the rate of change of angle with time)  must also be constant, hence

Thus, the observer, being in the same frame of reference, could describe the motion of the bike in two different ways, once using rectangular co-ordinate system and then using polar co-ordinate system. Both the descriptions are equivalent, but in our case polar co-ordinates are more convenient to use because it reduces the number of time dependent variables from two, x and y in xy co-ordinate system, to one θ in rθ co-ordinates system, thus simplifying the problem.

The two co-ordinate systems we used here can be transformed into one another using the following equations :

In general, it is always possible to take problem from one co-ordinate system to another where the problem may appear mathematically simpler. In practice, one adopts a co-ordinate system by looking at the symmetry involved in the given system. Our’s was a case of circular motion which looked much simpler in polar co-ordinates as it reduced the number of time dependent co-ordinates. Likewise, if the motion was restricted on the surface of a sphere, it would have been more convenient to use spherical co-ordinate systems (r, θ, ϕ) instead of rectangular co-ordinate system (x, y, z) as in spherical co-odinate system, we would have to deal with only two time dependent variables viz. θ and ϕ (r being constant throughout the motion). These kind of transformations are widely used in physics and are indispensable tools while solving  problems in higher dimensions which are mathematically much more intricate.