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Simple question on proving polar coordinates

  1. Dec 9, 2014 #1
    1. The problem statement, all variables and given/known data
    Prove: $$\frac{d\hat{r}}{dt} = \dot{\phi} \hat{\phi }$$ and $$\frac{d\hat{\phi}}{dt} = -\dot{\phi} \hat{r }$$


    2. Relevant equations


    3. The attempt at a solution
    I solved this for an Analytical Mechanics assignment a month ago, and completely forgot how it goes..
    $$\hat{r} ⊥ \hat{\phi}$$
    An change from r1 to r2 will create a ##Δ\phi## that is in the ##\hat{\phi}## direction...
    and because ##\hat{r} ⊥ \hat{\phi}##, we can say the same happens for a change from ##\phi1## to ##\phi2## except in the ##-\hat{r}## direction. Assuming the change is infinitesimal, we can write ##Δr## or ##Δ\phi## as d/dt.

    But then I'm confused because, why are we assuming a change from r1 to r2 is a rotation by ##Δ\phi,## and not a change of the length r..? Am I getting something completely wrong here?
     
  2. jcsd
  3. Dec 9, 2014 #2
    Go back to definition:
    ##\hat r = \cos(\phi(t)) \vec i + \sin(\phi(t)) \vec j ##
    ##\hat \phi = -\sin(\phi(t)) \vec i + \cos(\phi(t)) \vec j ##

    And use time derivative
     
  4. Dec 9, 2014 #3
    Go back to definition:
    ##\hat r = \cos(\phi(t)) \vec i + \sin(\phi(t)) \vec j ##
    ##\hat \phi = -\sin(\phi(t)) \vec i + \cos(\phi(t)) \vec j ##

    And use time derivative
     
  5. Dec 10, 2014 #4
    Thank you so much!
    My professor emphasized the geometric interpretation of the answers that I completely forgot about those definitions.
    Worked like magic, problem solved.

    By the way, is there a reason why you're writing ##\vec{i}## and not ##\hat{i}##?
    I'm used to ##\hat{i}## as a notation for unit vectors, but do you mean the same thing or are you referring to something else?
     
  6. Dec 10, 2014 #5
    Yes you're right, I meant ##\hat i## and ## \hat j ##.
    For clear explanations and nice drawings, look up Kleppner and Kolenkow first chapter.
     
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