Skip to main content
Physics

Return to Question

edited tags
Link
Qmechanic ♦
Qmechanic ♦
  • 212.9k
  • 48
  • 589
  • 2.3k
Post Closed as "Duplicate" by Qmechanic♦ classical-mechanics
edited body
Source Link
Cosmas Zachos
Cosmas Zachos
  • 66.3k
  • 6
  • 110
  • 248

The components of the classical angular momentum $L_i$, satisfy the Poisson bracket relation $$\{L_i,L_j\}=\epsilon_{ijk}L_k,\tag{1}$$ and forms a Lie algebra (i.e, anti-symmetric, obeys the Jacobi identity etc).

In the theory of Lie groups, the commutator $$[\hat{L}_i,\hat{L}_j]=i\epsilon_{ijk}\hat{L}_k\tag{1}$$$$[\hat{L}_i,\hat{L}_j]=i\epsilon_{ijk}\hat{L}_k\tag{2}$$ is definitely called a $\mathfrak{so}(3)$ Lie algebra, because the $\hat{L}_i$'s generate ${\rm SO}(3)$ group elements $\left(\exp\left(-i\theta_i\hat{L}_i\right)\right)$.

I wonder whether the PB Lie algebra $(1)$ is also called a $\mathfrak{so}(3)$ Lie algebra. And if so, why? What is the relation of $(1)$ to the ${\rm SO}(3)$ group, if any?

The components of the classical angular momentum $L_i$, satisfy the Poisson bracket relation $$\{L_i,L_j\}=\epsilon_{ijk}L_k,\tag{1}$$ and forms a Lie algebra (i.e, anti-symmetric, obeys the Jacobi identity etc).

In the theory of Lie groups, the commutator $$[\hat{L}_i,\hat{L}_j]=i\epsilon_{ijk}\hat{L}_k\tag{1}$$ is definitely called a $\mathfrak{so}(3)$ Lie algebra, because the $\hat{L}_i$'s generate ${\rm SO}(3)$ group elements $\left(\exp\left(-i\theta_i\hat{L}_i\right)\right)$.

I wonder whether the PB Lie algebra $(1)$ is also called a $\mathfrak{so}(3)$ Lie algebra. And if so, why? What is the relation of $(1)$ to the ${\rm SO}(3)$ group, if any?

The components of the classical angular momentum $L_i$, satisfy the Poisson bracket relation $$\{L_i,L_j\}=\epsilon_{ijk}L_k,\tag{1}$$ and forms a Lie algebra (i.e, anti-symmetric, obeys the Jacobi identity etc).

In the theory of Lie groups, the commutator $$[\hat{L}_i,\hat{L}_j]=i\epsilon_{ijk}\hat{L}_k\tag{2}$$ is definitely called a $\mathfrak{so}(3)$ Lie algebra, because the $\hat{L}_i$'s generate ${\rm SO}(3)$ group elements $\left(\exp\left(-i\theta_i\hat{L}_i\right)\right)$.

I wonder whether the PB Lie algebra $(1)$ is also called a $\mathfrak{so}(3)$ Lie algebra. And if so, why? What is the relation of $(1)$ to the ${\rm SO}(3)$ group, if any?

Source Link
SRS
SRS
  • 27.2k
  • 12
  • 106
  • 341

About the Lie algebra of the angular momentum Poisson bracket structure

The components of the classical angular momentum $L_i$, satisfy the Poisson bracket relation $$\{L_i,L_j\}=\epsilon_{ijk}L_k,\tag{1}$$ and forms a Lie algebra (i.e, anti-symmetric, obeys the Jacobi identity etc).

In the theory of Lie groups, the commutator $$[\hat{L}_i,\hat{L}_j]=i\epsilon_{ijk}\hat{L}_k\tag{1}$$ is definitely called a $\mathfrak{so}(3)$ Lie algebra, because the $\hat{L}_i$'s generate ${\rm SO}(3)$ group elements $\left(\exp\left(-i\theta_i\hat{L}_i\right)\right)$.

I wonder whether the PB Lie algebra $(1)$ is also called a $\mathfrak{so}(3)$ Lie algebra. And if so, why? What is the relation of $(1)$ to the ${\rm SO}(3)$ group, if any?