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I Proof of angular momentum conservation

  1. Aug 6, 2016 #1
    upload.png
    This is from text [Introduction to Lagrangian and Hamiltonian Mechanics] on NTNU opencourse.
    Annnnd... I don't use english as my primary language, so sorry for poor sentences.

    I can't get two things in here.

    First, at (1.12) I can't understand how L dot derivated like that.
    Since I know differentiation of cross product should be done like

    d/dt(AxB)=d/dt(A) x B + A x d/dt(B)

    then, at (1.12), why it doesn't have the terms of d/dt(r) x p ?
    I think it only has the terms of r x d/dt(p)

    Second, I can't get how
    upload2.png
    were derived by using (1.13), How could ri X Fji = 1/2(rij X Fji)
    is possible?

    These might be dumb questions, but please help me.
     
  2. jcsd
  3. Aug 6, 2016 #2
    ##\boldsymbol p_i=m_i\dot{\boldsymbol r}_i,\quad \dot{\boldsymbol r}_i\times\dot{\boldsymbol r}_i=0##
    by the way ##\sum_{ij}{\boldsymbol r}_{i}\times\boldsymbol F_{ji}=0##
     
  4. Aug 6, 2016 #3
    Thank you for clear explanation. Got the first one.

    Yes Σij rij x Fji = 0 so eventually it makes L dot = tau (torque)
    But, what I want know is the mathematical manuever that makes ri X Fji = 1/2(rij X Fji)
     
  5. Aug 6, 2016 #4
    do the calculation for two particles directly and everything will be clear
     
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