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Vector calculus question

  1. Oct 17, 2014 #1
    Problem: Consider a system for which Newton's second law is $$ \frac {d \vec v}{dt} = - [ \frac {h(r)h'(r)}{r} + \frac {k}{r^3} ] \vec r- \frac {h'(r)}{r} \vec L $$ where k is a constant, h(r) is some function of r, h'(r) is its derivative and L = r x v is the angular momentum. Show that $$ \frac {d \vec L}{dt} = - \frac {h'(r)}{r} \vec r × \vec L$$ and use this equation to prove that L is not generally conserved, but its magnitude L is conserved.

    Attempt: I've done the first part of the question, but I don't know how I should go about showing that L is not conserved but its magnitude is conserved. Any hints would be appreciated.
     
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  3. Oct 17, 2014 #2

    Dick

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    The magnitude squared is given by the dot product of L with itself. Can you show the time derivative of that is 0?
     
  4. Oct 17, 2014 #3
    So |L|^2 = (r×v)⋅(r×v) = (rr)(vv) - (vr)(vr) = |r|^2 |v|^2 right? But how would I show the time derivative of this to be 0? [itex] \frac {dL}{dt} = r \frac {dv}{dt} + v \frac {dr}{dt} [/itex], but how do I make this equal 0?
     
  5. Oct 17, 2014 #4

    Dick

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    You want to show the time derivative of ##L \cdot L## is zero. Use the product rule and your given expression for dL/dt. Can you tell me why dL/dt must be perpendicular to L?
     
    Last edited: Oct 17, 2014
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