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Goldstein Poole Schmidt chapter 4, Exercises of Classical Mechanics

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 0-201-65702-3
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).
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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.

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

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)

  1. Commutation relations for angular momentum, 15 points.

(a) Problem 4.17. Verify commutation relations for the generators of rotation

M

i

in GPS Eqs. (4.79) and (4.80). Here I use the hat to indicate that

M

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

L

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.

  1. Inspired by Problem 4.10, 10 points. Don’t do Problem 4.10; only answer the

questions below.

Let us denote by

R

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:

R

z

(φ + dφ) =

R

z

(dφ)

R

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 →

R.

(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)

Mz

Rz (φ). (7)

Show that the solution of this equation is

R

z

(φ) = e

ˆ Mz φ

=

∞ ∑

n=

M

z

φ)

n

n!

(b) Let us denote by ˆm z

(

)

the (x, y) part of the matrix

M

z

in Eq. (4.79).

(The third column and row of

Mz are trivial and decouple from the first two.)

Show that

2

z

1 , mˆ

3

z

= − mˆ z

, mˆ

4

z

(c) Using Eq. (9) in Eq. (8), show that

R

z

(φ) =

cos φ − sin φ 0

sin φ cos φ 0

 ,^ (10)

in agreement with GPS Eq. (4.43). (Never mind the difference in the sign of φ.)

October 20, 2014