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Time derivatives in Spherical Polar Coordinates

  1. Aug 25, 2015 #1
    1. The problem statement, all variables and given/known data
    Evaluate r(hat and overdot), θ(hat and overdot), φ(hat and overdot) in terms of (θ , Φ) and the time derivatives of the two remaining spherical polar coordinates. Your results should depend on the spherical polar unit vectors.

    2. Relevant equations

    ∂/∂t=

    3. The attempt at a solution

    I just need some help understanding what is being asked here. The professor is no help when it comes to asking him for help understanding what he wants. It sounds like he wants me to evaluate the time derivatives of the unit vectors in spherical polar coordinates in relation to the angles. I don't know what he means by the "other two" spherical polar coordinates. Any insight would be helpful!

    Thanks!
     
  2. jcsd
  3. Aug 25, 2015 #2
    Given a vector
     
  4. Aug 25, 2015 #3
    Given a vector
    v= {vr,vθ,vφ}
    you can write this in terms of the unit vectors
    = vr er+ vθ eθ+ vφ eφ.
    There I am using the notation that ei is the unit vector associated in the i'th direction.

    If you want to take the time derivative of this vector
    t v = ∂t (vr er+ vθ eθ+ vφ eφ)
    =∂t (vr er)+ ∂t (vθ eθ)+∂t (vφ eφ).

    Each component then has a product rule associated with it. For the r direction
    t (vr er= ∂t (vr) er)+ vrt (er)

    and from there it is necessary to figure out how the unit vectors change in time. In Cartesian co-ordinates they do not, but this is no longer the case in polar co-ordinates.

    For reference, if you have or can find a copy of, Taylor Classical Mechanics does this for planar polar systems. The expansion into spherical polar is more tedious but simply a continuation.
     
  5. Aug 26, 2015 #4
    Suppose you have a function f=f(x,y), where x and y are represented as parametric functions of time t. How would you find the time derivative of f, df/dt?
     
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