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


by yicong2011
Tags: angular momentum, components, independent
yicong2011
yicong2011 is offline
#1
Apr23-11, 08:35 PM
P: 76
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.
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RedX
RedX is offline
#2
Apr23-11, 10:41 PM
P: 969
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.
Bill_K
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#3
Apr24-11, 03:49 PM
Sci Advisor
Thanks
Bill_K's Avatar
P: 3,853
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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