### Choosing a new coordinate system

I often used to wonder why one coordinate system is more appropriate than another in particular situations.

For example, when dealing with circular motion, we are often advised to use polar coordinates. I used to wonder why we can’t use only $x-y$ coordinates. My teacher used to say using polar coordinates makes things easier. But I never quite bought it. Although after working on such situations with $x-y$ coordinates I did realise that formulae were much simpler with polar coordinates, I didn’t think this justified completely changing the coordinate system.

This I feel is something many people can relate to.

The reason why changing the coordinate system helps is that if one can find axes that perfectly align with the motion of the object, then equations become much simpler! Let me generalize this, for a better insight.

Let us suppose an object can move only in a wave like fashion, and that too only in one direction. Normally, one may take the $x-y$ axis, and then represent the location of the object through a complicated formula. However, what if we design an axis similar in shape to the wave?! We can then represent the location only through one coordinate!

This is true for any kind of motion. However, at this point, one may say that if *any* motion can be so simply represented, why don’t we always take such an axis? This is because choosing a new axis and hence simplifying representation does not solve all problems. You also have to eventually convert to the coordinate system you’re working with. Hence, if the cost of converting back to the *home* coordinate system is balanced out by the benefits availed by choosing a new axis, then go for it! Else, don’t.