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

[yicong2011]

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

L_x, L_y, L_z

independent to each other?

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

[L_i, L_j] = ε_ijk L_k.

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

If the identity is true, then the three components of Angular Momentum are not independent to each other.

 

[RedX]

Angular momenta are independent of each other. Take the obvious case of planar motion in the xy-plane. Then L_x and L_y are zero, and L_z can be anything.

 

[Bill_K]

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