Are Lx, Ly, Lz (the components of Angular Momentum)independent to each other?

  1. In Classical Mechanics, are the three components of Angular Momentum L:

    Lx, Ly, Lz

    independent to each other?

    It seems that there is an identity in Classical Mechanics (Sorry, I can hardly remember where I saw it):

    [Li, Lj] = εijk Lk.

    Note: [] is Poisson Bracket, εijk is Levi-Civita Tensor

    If the identity is true, then the three components of Angular Momentum are not independent to each other.
     
  2. jcsd
  3. Angular momenta are independent of each other. Take the obvious case of planar motion in the xy-plane. Then Lx and Ly are zero, and Lz can be anything.
     
  4. Bill_K

    Bill_K 4,159
    Science Advisor

    yicong, Of course you are misinterpreting the meaning of the Poisson bracket. Note for example in three dimensions with Cartesian coordinates xi, the Poisson bracket relationship [xi, xj] = 0 for i and j not equal.
     
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