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.