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Self explanatory title. This is a collection of problems on chapter 4 from golstein's classical mechanics book.
Typology: Exercises
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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-
LL: Landau and Lifshitz, Mechanics, 3rd edition, 1976, ISBN 978-0-7506-2896-
Total score is 40 points.
Do Part (b) first, because it is easier. Additional question to Part (b): Show that
ijk
ijk
Then do Part (a). Additional questions: Using the formula derived in Part (a),
prove the following vector relations
(a) Problem 4.17. Verify commutation relations for the generators of rotation
i
in GPS Eqs. (4.79) and (4.80). Here I use the hat to indicate that
i
is a matrix,
whereas the index i indicates 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 f and g is defined in LL Eq. (42.5) and
GPS Eq. (9.67) as
[f, g] =
∂f
∂r k
∂g
∂p k
∂f
∂p k
∂g
∂r k
where summation of the repeated index k is implied in tensor notation.
(b) Let us introduce the angular momentum L i
= (r × p) i
ijk
r j
p k
, the infinites-
imal angles of rotation δφ, and the scalar product δφ · L = δφ j
j
. 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, δφ · L] = p × δφ. (5)
Here the angles δφ are constant parameters, which do not depend on r and p, so
they are not subject to differentiation in the Poisson brackets. Thus, the angular
momentum L is the generator of rotations via the Poisson brackets in Eq. (5).
2 Homework #6, Phys601, Fall 2014, Prof. Yakovenko
(c) Calculate the Poisson brackets between different components of the angular mo-
mentum L = r × p and prove the following relation
[Li, Lj ] = ijkLk. (6)
Compare Eq. (6) with the commutation relations of the angular momentum op-
erator in quantum mechanics.
questions below.
Let us denote by
z
(φ) the matrix of rotation around the axis z by the angle φ. Then,
a rotation by φ + dφ is a composition of two rotations:
z
(φ + dφ) =
z
(dφ)
z
(φ),
where
Rz (dφ) = ˆ1 +
Mz dφ. Here I used the equation above GPS Eq. (4.79) and the
equation above GPS Eq. (4.68) with the change of notation A →
(a) Introducing the notation d
Rz (φ) =
Rz (φ + dφ) −
Rz (φ), show that the rotation
matrix satisfies the following differential equation:
d
Rz (φ)
dφ
Mz
Rz (φ). (7)
Show that the solution of this equation is
z
(φ) = e
ˆ Mz φ
=
∞ ∑
n=
z
φ)
n
n!
(b) Let us denote by ˆm z
(
)
the (x, y) part of the matrix
z
in Eq. (4.79).
(The third column and row of
Mz are trivial and decouple from the first two.)
Show that
mˆ
2
z
1 , mˆ
3
z
= − mˆ z
, mˆ
4
z
(c) Using Eq. (9) in Eq. (8), show that
z
(φ) =
cos φ − sin φ 0
sin φ cos φ 0
in agreement with GPS Eq. (4.43). (Never mind the difference in the sign of φ.)
October 20, 2014