Homework #6 — PHYS 601 — Fall 2014
Due on Thursday, October 27, 2014 online
Professor Victor Yakovenko
Office: 2115 Physics
Web page: http://physics.umd.edu/~yakovenk/teaching/
GPS: Goldstein, Poole, Safko, Classical Mechanics, 3rd edition, 2002, ISBN 0-201-65702-3
LL: Landau and Lifshitz, Mechanics, 3rd edition, 1976, ISBN 978-0-7506-2896-9
Total score is 40 points.
The Kinematics of Rigid Body Motion, GPS Ch. 4
1. Problem GPS 4.14, 15 points. Identities for antisymmetric tensors.
Do Part (b) first, because it is easier. Additional question to Part (b): Show that
ijk ijk = 6.
Then do Part (a). Additional questions: Using the formula derived in Part (a),
prove the following vector relations
(A×B)·(C×D)=(A·C) (B·D)−(A·D) (B·C) (1)
A×(B×C) = B(A·C)−C(A·B) (2)
(A×B)×(C×D) = [A·(B×D)] C−[A·(B×C)] D(3)
2. Commutation relations for angular momentum, 15 points.
(a) Problem 4.17. Verify commutation relations for the generators of rotation ˆ
Mi
in GPS Eqs. (4.79) and (4.80). Here I use the hat to indicate that ˆ
Miis a matrix,
whereas the index iindicates that there are three such matrices. Compare with the
commutation relations of the angular momentum operator in quantum mechanics.
Additional questions:
The Poisson bracket [f, g] of two functions fand gis defined in LL Eq. (42.5) and
GPS Eq. (9.67) as
[f, g ] = ∂f
∂rk
∂g
∂pk
−∂f
∂pk
∂g
∂rk
,(4)
where summation of the repeated index kis implied in tensor notation.
(b) Let us introduce the angular momentum Li= (r×p)i=ijkrjpk, the infinites-
imal angles of rotation δφ, and the scalar product δφ·L=δφjLj. Calculate
explicitly the Poisson brackets in the following equations in tensor notation and
show that they give the right-hand sides of these equations, in agreement with
GPS Eq. (4.75)
δr= [r, δφ·L] = r×δφ, δp= [p, δ φ·L] = p×δφ.(5)
Here the angles δφare constant parameters, which do not depend on rand p, so
they are not subject to differentiation in the Poisson brackets. Thus, the angular
momentum Lis the generator of rotations via the Poisson brackets in Eq. (5).
