The discussion revolves around the use of polar coordinates in one-dimensional problems, specifically focusing on a particle constrained to move in the y-direction while its x-coordinate remains fixed. Participants explore whether expressing y in terms of polar coordinates complicates the problem unnecessarily.
Participants generally agree that using polar coordinates in this context may complicate the problem, but there is no consensus on the overall utility or appropriateness of such a choice.
The discussion highlights the dependence on the choice of coordinate systems and the implications of fixing one coordinate while varying another, but does not resolve the broader question of when polar coordinates are advantageous.
Yes.kent davidge said:so this has only made solving the problem harder than it must be, right?
If you do this, only one of your polar coordinates will be a free parameter as the other will be fixed. For example, consider the line ##x = 1##. On this line, ##r^2 = 1 + y^2## which means that, if you fix ##\theta##, then ##r^2(1-\sin^2\theta) = 1## and so ##r = 1/\cos\theta##. The relation between ##y## and ##\theta## is therefore ##y = \tan\theta##, which is just a (non-linear) change of parametrisation of your line.kent davidge said:If I have a physical problem, say, a particle which is constrained to move in the ##y## direction, which means that its ##x## coordinate remains fixed, does it make sense to write ##y## in terms of polar coordinates? That is, ##y = r \sin\theta##. Since now I have two parameters ##r,\theta## varying, so this has only made solving the problem harder than it must be, right?