Question

What is the poisson bracket of momentum and angular momentum?

Answer 1

I'm trying to find [Mi,Mj][Mi,Mj] Poisson brackets.

{Mi,Mj}=∑l(∂Mi∂ql∂Mj∂pl−∂Mi∂pl∂Mj∂ql){Mi,Mj}=∑l(∂Mi∂ql∂Mj∂pl−∂Mi∂pl∂Mj∂ql)

I know that:

Mi=ϵijkqjpkMi=ϵijkqjpk

Mj=ϵjnmqnpmMj=ϵjnmqnpm

and so:

[Mi,Mj]=∑l(∂ϵijkqjpk∂ql∂ϵjnmqnpm∂pl−∂ϵijkqjpk∂pl∂ϵjnmqnpm∂ql)[Mi,Mj]=∑l(∂ϵijkqjpk∂ql∂ϵjnmqnpm∂pl−∂ϵijkqjpk∂pl∂ϵjnmqnpm∂ql)

=∑lϵijkpkδjl⋅ϵjnmqnδml−∑lϵijkqjδkl⋅ϵjnmpmδnl=∑lϵijkpkδjl⋅ϵjnmqnδml−∑lϵijkqjδkl⋅ϵjnmpmδnl

Then I have thought that values that nullify deltas don't add any informations in the summations. And so, m=l,j=lm=l,j=l but so I obtain m=jm=j. But ifm=lm=l, the second Levi-Civita symbol in the first summation is zero... And if I go on, I obtain {Mi,Mj}=−piqj{Mi,Mj}=−piqj inste

Answer 2

Answer:

Poisson brackets are another formal formulation of classical mechanics. They help make the connection between symmetries and conservation laws more explicit. The Poisson bracket of the x,y,z components of angular momentum are derived. ... Poisson brackets of the x, y, z components of angular momentum.