- #1

- 10

- 0

## Homework Statement

Let f(q, p), g(q, p) and h(q, p) be three functions in phase space. Let Lk =

ε

_{lmk}q

_{l}p

_{m}be the kth component of the angular momentum.

(i) Define the Poisson bracket [f, g].

(ii) Show [fg, h] = f[g, h] + [f, h]g.

(iii) Find [q

_{j}, L

_{k}], expressing your answer in terms of the permutation symbol.

(iv) Show [L

_{j}, L

_{k}] = q

_{j}p

_{k}−q

_{k}p

_{j }. Show also that the RHS satisfies q

_{j}p

_{k}−q

_{k}pj =

ε

_{ijk}L

_{i}. Deduce [L

_{i}, |

**L**|

^{2}] = 0.

[Hint: the identity ε

_{ijk}ε

_{klm}= δ

_{il}δ

_{jm}− δ

_{im}δ

_{jl}may be useful in (iv)]

## Homework Equations

n/a## The Attempt at a Solution

i) [f,g]=[itex]\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}[/itex]

ii) easy to show from the definition in i)

iii) after a bit of working, I get ε

_{lmk}q

_{l}

iv) my working is quite long, but I get [L

_{j},L

_{k}]=q

_{j}p

_{k}-q

_{k}p

_{j}=ε

_{ijk}L

_{i}as required.

The bit I'm having trouble with is the very last bit of the question, to deduce [L

_{i}, |

**L**|

^{2}] = 0.

Since it's only a small part of the question, it seems as though this part should be fairly simple so maybe I'm overlooking something, but I don't get 0. This is my working:

[L

_{i}, |

**L**|

^{2}]=[L

_{i}, L

_{j}L

_{j}]=L

_{j}[L

_{i}, L

_{j}]+[L

_{i}, L

_{j}]L

_{j}=2L

_{j}[L

_{i}, L

_{j}]

I'm not entirely sure where to go from here so any help (or pointing out of any glaring errors) would be great.