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Momentum for my exam tomorrow

  1. Apr 15, 2008 #1
    1. The problem statement, all variables and given/known data
    Let P be the 4 momentum
    u be the 4 velocity
    a) Evaluate the Lorentz invariant [itex]P^2[/itex]
    b)Differentiate [itex]P^2=P_{\mu}P_{\mu}[/itex] and show that
    [tex]\vec{u}\cdot\frac{d}{dt}\left(\frac{m_{0}\vec{u}}{\sqrt{1-u^2/c^2}}\right)=m_{0}c^2\frac{d}{dt}\left(\frac{1}{\sqrt{1-u^2/c^2}}\right)[/tex]

    2. The attempt at a solution

    The first part yields an answer of [tex]E^2 -p^2 c^2=m^2 c^4[/tex]

    Now for part b. Does the P^2 have anything to do with the equality that needs to be proven? Do i need to differentiate P^2 with respect to time? Do i hav to differentiate
    [tex]E^2 -p^2 c^2=m^2 c^4[/tex] with respect to time?

    Please help!

    Thanks
     
  2. jcsd
  3. Apr 15, 2008 #2

    nrqed

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    Yes, differentiate with respect to time. The derivative of P^2 obviously gives zero since m^2 c^4 is constant.

    Now, write P^2 = E^2 - p dot p c^2

    (my small p is the three momentum and dot is the dot product

    so d/dt(P^2) = 2 E dE/dt - 2 c^2 p dot dp/dt

    This must be zero. Now write E= gamma mc^2 and p = gamma m u where u is the ordinary three velocity
     
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