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

  • Thread starter Thread starter yicong2011
  • Start date Start date
  • Tags Tags
    Angular Components
AI Thread Summary
The discussion centers on the independence of the components of Angular Momentum (Lx, Ly, Lz) in Classical Mechanics. It references the Poisson bracket identity, suggesting that if true, the components are not independent. However, a counterargument highlights that in specific cases, such as planar motion, Lx and Ly can be zero while Lz remains variable, indicating a form of independence. The interpretation of the Poisson bracket is also debated, with examples provided to clarify its implications. Overall, the relationship between the components of Angular Momentum remains a nuanced topic in classical physics.
yicong2011
Messages
75
Reaction score
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.
 
Physics news on Phys.org
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.
 
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.
 

Thread 'Derivation of Hamilton's Principle: Questions'

I have recently been really interested in the derivation of Hamiltons Principle. On my research I found that with the term ##m \cdot \frac{d}{dt} (\frac{dr}{dt} \cdot \delta r) = 0## (1) one may derivate ##\delta \int (T - V) dt = 0## (2). The derivation itself I understood quiet good, but what I don't understand is where the equation (1) came from, because in my research it was just given and not derived from anywhere. Does anybody know where (1) comes from or why from it the...
View full post »

Similar threads

I Why “If Lz has a well-defined value, then Lx and Ly do not”？
Replies
6
Views
3K
Angular momentum L=rxP and L=I x omega ?
Replies
1
Views
5K
I Question about Stern-Gerlach experiment
Replies
12
Views
3K
What is quantum angular momentum
Replies
1
Views
3K
Conservation of angular momentum (central force)
Replies
3
Views
2K
Solving for <Lx^2> and <Ly^2>: How do I show that <Lx^2> = <Ly^2>?
Replies
12
Views
12K
Angular Momentum Term Equals Zero?
Replies
3
Views
1K
What is the disparity in the dimensionality of angular momenta?
Replies
8
Views
3K
Common sets of eigenfunctions in angular momentum
Replies
4
Views
4K
Problem with Poisson's Theorem
Replies
5
Views
6K

Hot Threads

Recent Insights

Back
Top