ADVANCED DYNAMICS — PHY 4241/5227 HOME AND CLASS WORK – SET 4 (January 26, 2011)

Read Landau-Lifshitz §10 and §11. Due January 28. (14a) Use the Einstein summation convention in the following. Show that

L_{i} = ϵ_{ijk} x_{j} p_{k} = [r⃗ × p⃗]_{i} holds for i = 1, 2, 3.

Due in class (2 points). (14b) Express ϵ_{12k} ϵ_{lmk} in terms of Kronecker delta and then ϵ_{ijk} ϵ_{lmk}. Due in class

(2 points).

(14c) Poisson brackets are defined by

[g, h] = ∑_{k} (\frac{∂g}{∂q_{k}} \frac{∂h}{∂p_{k}} − \frac{∂h}{∂q_{k}} \frac{∂g}{∂p_{k}} )

where g and h are functions of q_{i}, p_{i} and, possibly, t. Show the following properties (due February 2 before class, 10 points):

1.

\frac{dg}{dt} = [g, H] + \frac{∂g}{∂t} .

2.

q̇_{j} = [q_{j}, H] .

3.

ṗ_{j} = [p_{j}, H] .

4.

[x_{i}, x_{j}] = [p_{i}, p_{j}] = 0 ; [x_{i}, p_{j}] = δ_{ij} ,

5.

[x_{i}, L_{j}] = ϵ_{ijk}x_{k} , [p_{i}, L_{j}] = ϵ_{ijk}p_{k} , and [L_{i}, L_{j}] = ϵ_{ijk}L_{k} , ,

where the Einstein summation convention is used and L_{j} = ϵ_{jkl} x_{k} p_{l} is the jth component of the angular momentum of the system.

Landau-Lifshitz: Skip §12, read §13 (due February 2) and §14 (due February 4).