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Poisson brackets little problem

  1. Aug 6, 2011 #1

    fluidistic

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    Gold Member

    1. The problem statement, all variables and given/known data
    For a particle, calculate Poisson brackets formed by:
    1)The Cartesian components of the linear momentum [itex]\vec p[/itex] and the angular momentum [/itex]\vec M =\vec r \times \vec p[/itex].
    2)The Cartesian components of the angular momentum.


    2. Relevant equations

    [itex][u,p]_{q,p}= \sum _k \left ( \frac{\partial q }{\partial q_k } \frac{\partial v }{\partial p _k} -\frac{\partial q }{\partial p_k } \frac{\partial v }{\partial q _k} \right )[/itex].

    3. The attempt at a solution
    2)Nothing still, waiting to complete 1).
    1)I calculated the Cartesian components of M and p.
    I don't understand what I have to calculate. [itex][\vec p, \vec M][/itex] I'm guessing but with what subscript?
    Thanks for any help.

    Edit: Hmm I think the subscript is always q,p. But p and M are vectors, so have I to calculate directional derivative?
     
    Last edited: Aug 6, 2011
  2. jcsd
  3. Aug 6, 2011 #2
    [tex] \vec{p} = ( p_x , p_y , p_z ) = p_i [/tex] and [tex] \vec{r} = ( x , y , z ) = x_i [/tex]

    [tex] M_i = \epsilon_{ijk} x_j p_k [/tex] where the einstein summation convention is used

    [tex] [p_i,M_j]= \sum _l \left ( \frac{\partial p_i }{\partial q_l } \frac{\partial M_j }{\partial p _l} -\frac{\partial p_i }{\partial p_l } \frac{\partial M_j }{\partial q _l} \right ) [/tex]

    hope this helps
     
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