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

In summary, the three components of Angular Momentum in Classical Mechanics - Lx, Ly, and Lz - are independent of each other. This is evident in the identity [Li, Lj] = εijk Lk, where [] is the Poisson Bracket and εijk is the Levi-Civita Tensor. This means that even in the case of planar motion, where Lx and Ly are zero, Lz can still have a non-zero value. Therefore, the three components are not dependent on each other, contrary to the belief of the person mentioned in the conversation.
  • #1
yicong2011
75
0
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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  • #2
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.
 
  • #3
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.
 

1. Are Lx, Ly, and Lz always independent from each other?

No, Lx, Ly, and Lz are not always independent from each other. In certain situations, such as when dealing with a symmetric system, these components may be dependent on each other.

2. How do changes in Lx affect Ly and Lz?

Changes in Lx can affect Ly and Lz in different ways. For example, if Lx is increased, Ly and Lz may remain constant or they may also increase or decrease depending on the system.

3. Can one component of angular momentum be zero while the others are non-zero?

Yes, it is possible for one component of angular momentum to be zero while the others are non-zero. This is known as partial angular momentum and can occur in systems with certain symmetries.

4. Are there any physical quantities that are always conserved for each component of angular momentum?

Yes, for each component of angular momentum, there are corresponding physical quantities that are always conserved. For example, Lx is associated with the conservation of linear momentum in the x-direction.

5. How do Lx, Ly, and Lz combine to give the total angular momentum?

The total angular momentum, denoted as L, is the vector sum of all three components: L = Lx + Ly + Lz. This means that the direction and magnitude of the total angular momentum is determined by the combination of all three components.

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