PHY 711 Classical mechanics and mathematical methods

PHY 711 Classical Mechanics and Mathematical Methods

MWF 10 AM-10:50 PM

OPL 103

http://www.wfu.edu/~natalie/f14phy711/

Instructor: Natalie Holzwarth Phone:758-5510 Office:300 OPL e-mail:natalie@wfu.edu

Course schedule

(Preliminary schedule -- subject to frequent adjustment.) rrr>

	Date	F&W Reading	Topic	Assignment
1	Wed, 8/27/2014	Chap. 1	Review of basic principles	#1
2	Fri, 8/29/2014	Chap. 1	Scattering theory	#2
3	Mon, 9/01/2014	Chap. 1	Scattering theory continued	#3
4	Wed, 9/03/2014	Chap. 2	Accelerated coordinate systems	#4
5	Fri, 9/05/2014	Chap. 3	Calculus of variations	#5
6	Mon, 9/08/2014	Chap. 3	Calculus of variations	#6
7	Wed, 9/10/2014	Chap. 3	Hamilton's principle	#7
8	Fri, 9/12/2014	Chap. 3 & 6	Hamilton's principle	#8
9	Mon, 9/15/2014	Chap. 3 & 6	Lagrangians with constraints	#9
10	Wed, 9/17/2014	Chap. 3 & 6	Lagrangians and constants of motion	#10
11	Fri, 9/19/2014	Chap. 3 & 6	Hamiltonian formalism	#11
12	Mon, 9/22/2014	Chap. 3 & 6	Hamiltonian formalism	#11
13	Wed, 9/24/2014	Chap. 3 & 6	Hamiltonian Jacobi transformations
14	Fri, 9/26/2014	Chap. 4	Small oscillations	Begin Take-Home
15	Mon, 9/29/2014	Chap. 4	Normal modes of motion	Continue Take-Home
16	Wed, 10/01/2014	Chap. 4	Normal modes of motion	Continue Take-Home
17	Fri, 10/03/2014	Chap. 4	Normal modes of motion	Take-Home due
18	Mon, 10/06/2014	Chap. 7	Wave motion	#12
19	Wed, 10/08/2014	Chap. 7	Sturm-Liouville Equations	#13
20	Fri, 10/10/2014	Chap. 7	Sturm-Liouville Equations	#13
21	Mon, 10/13/2014	Chap. 7	Sturm-Liouville Equations	#14
22	Wed, 10/15/2014	Appendix A	Contour integration methods	#15
	Fri, 10/17/2014		Fall break -- no class
23	Mon, 10/20/2014	Appendix A	Fourier transforms	#16
24	Wed, 10/22/2014	Chap. 5	Motion of Rigid Bodies	#17
25	Fri, 10/24/2014	Chap. 5	Motion of Rigid Bodies	#18
26	Mon, 10/27/2014	Chap. 5	Symmetric top in gravitational field	#18
27	Wed, 10/29/2014	Chap. 8	Vibrations of membranes	#19
28	Fri, 10/31/2014	Chap. 9	Physics of fluids	#20
29	Mon, 11/03/2014	Chap. 9	Physics of fluids	#21
30	Wed, 11/05/2014	Chap. 9	Sound waves
31	Fri, 11/07/2014	Chap. 9	Sound waves	Begin Take-Home
32	Mon, 11/10/2014	Chap. 9	Non-linear effects	Continue Take-Home
33	Wed, 11/12/2014	Chap. 10	Surface waves in fluids	Continue Take-Home
34	Fri, 11/14/2014	Chap. 10	Surface waves in fluids	Continue Take-Home
35	Mon, 11/17/2014	Chap. 11	Heat Conduction	Take-Home due #22
36	Wed, 11/19/2014	Chap. 12	Viscosity	#23
37	Fri, 11/21/2014	Chap. 12	More viscosity	#24
38	Mon, 11/24/2014	Chap. 13	Elastic Continua	Prepare presentations
	Wed, 11/26/2014		Thanksgiving Holiday
	Fri, 11/28/2014		Thanksgiving Holiday
39	Mon, 12/01/2014	Chap. 13	Elastic Continua	Prepare presentations
	Wed, 12/03/2014		Student presentations I
	Fri, 12/05/2014		Student presentations II
	Mon, 12/08/2014		Begin Take-home final

No Title PHY 711 - Assignment #1

08/27/2014

PDF version

Use maple or mathematica to plot the function

f(x)=e^−x²

and to evaluate the integral

⌠
⌡ 5

0
f(x) dx.

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On 26 Aug 2014, 23:34.

No Title PHY 711 - Assignment #2

08/29/2014

PDF version

In evaluating the differential cross section for Rutherford scattering, it is necessary to evaluate the following relationship involving the scattering angle θ, the impact parameter b, and a length parameter κ which involves the ratio of the interaction strength to the system energy:

π
2
− θ
2
= ⌠
⌡ ∞

κ+√{κ²+b²}
b
r
1

√

r²−2 κr −b²

dr.

Use Maple or other algebraic manipulation software to evaluate the integral to show that

2b= κ
tan(θ/2)
.
From this form of the impact parameter b(θ), "derive" the Rutherford scattering cross section.

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On 1 Sep 2014, 11:35.

No Title PHY 711 - Assignment #3

9/1/2014

PDF version

In class, we showed that the relationship between the impact parameter b and the scattering angle χ for elastic scattering between two hard spheres has the form:

b = D cos ⎛
⎝ χ
2
⎞
⎠ .

Using the above diagram which shows the geometry of two hard spheres at the moment of impact, derive this formula and the corresponding differential scattering cross section.

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On 31 Aug 2014, 21:19.

PHY 711 -- Assignment #4

Sept. 3, 2014

Start reading Chapter 2 in Fetter & Walecka. The following problem concerns material from Chapter 1.

Figure 5.3 of your text shows the relationship between the scattering angle θ and maximum orbital angle φ_m. In class we explicitly considered the repulsive Coulomb interaction (5.3b) with γ >0. Figure (5.3a) shows the situation when there is an attractive Coulomb interaction (such as when the incident and target particles have opposite signs of charge or such as in a gravitational interaction). Convince yourself that the form of the cross section (in the center of mass frame) remains the same if γ<0.

PHY 711 -- Assignment #5

Sept. 3, 2014

Start reading Chapter 3 in Fetter & Walecka.

Using calculus of variations, find the equation of the shortest "curve" which passes through the points (x=0, y=0) and (x=2, y=4).

No Title PHY 711 - Assignment #6

Sep 7, 2014

PDF version

This exercise is designed to illustrate the differences between partial and total derivatives.

Consider an arbitrary function of the form f=f(q,· q,t), where it is assumed that q=q(t) and · q ≡ dq/dt.
1. Evaluate
  
  ∂
  ∂q
  df
  dt
  − d
  dt
  ∂f
  ∂q
  .
2. Evaluate
  
  ∂
  ∂
  ⋅
  
  q
  
  df
  dt
  − d
  dt
  ∂f
  ∂
  ⋅
  
  q
  
  .
3. Evaluate
  
  df
  dt
  
  .
4. Now suppose that
  
  f(q,
  ⋅
  
  q
  
  ,t) = q
  ⋅
  
  q
  
  t, where q(t)=e^−t/τ.
  
  Here τ is a constant. Evaluate df/dt using the expression you just derived. Now find f(t) and take its time derivative directly to check your previous results.

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On 7 Sep 2014, 23:03.

PHY 711 -- Assignment #7

Sept. 10, 2014

Continue reading Chapter 3 in Fetter & Walecka.

Work Problem 3.10 (a) and (b); extra credit for (c), found at the end of Chapter 3 in your textbook.

PHY 711 -- Assignment #8

Sept. 12, 2014

Continue reading Chapter 3 and also 6 in Fetter & Walecka.

Consider a particle of mass m and charge q moving in constant magnetic field as described by the vector potential A(r) = B₀x u_y. Here u_y denotes a unit vector in the y direction. Find the Lagrangian and the equations of motion for this case. Compare with similar results discussed in the lecture notes.

No Title PHY 711 - Assignment #9

9/15/2014

PDF version

Continue reading Chapters 3 and 6 in Fetter and Walecka.

The figure above shows a box of mass im sliding on the frictionless surface of an inclined plane (angle θ). The inclined plane itself has a mass M and is supported on a horizontal frictionless surface. Write down the Lagrangian for this system in terms of the generalized coordinates X and s and solve for the equations of motion, assuming that the system is initially at rest.

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On 13 Sep 2014, 18:59.

No Title PHY 711 - Assignment #10

9/17/2014

PDF version

Continue reading Chapters 3 and 6 in Fetter and Walecka.

Consider the Lagrangian:

L(α,β,γ,
⋅

α

,
⋅

β

,
⋅

γ

) = A ⎛
⎝
⋅

α

2

sin² β+
⋅

β

2

⎞
⎠ + B ⎛
⎝
⋅

α

cosβ+
⋅

γ

⎞
⎠ 2

− C cosβ.

In this expression, A, B, and C represent given constant parameters. [You may (and will later) recognize this Lagrangian from the motion of a symmetric top.]
Find three constants of motion for this system. (It is not necessary to simplify the expressions.)

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On 19 Sep 2014, 09:20.

PHY 711 -- Assignment #11

Sept. 19, 2014

Continue reading Chapter 3 and also 6 in Fetter & Walecka.

Work problem 6.5 at the end of Chapter 6 of Fetter & Walecka first using Newtonian analysis and then by using Lagrangian or Hamiltonian formalisms, presumably getting the same result.

PHY 711 -- Assignment #12

Oct. 6, 2014

Finish reading Chapter 4 and start Chapter 7 in Fetter & Walecka.

Consider the system of 3 masses (m₁=m₂=m₃=m) shown attached by elastic forces in the right triangular configuration (with angles 45, 90, 45 deg) shown above with spring constants k and k'. Find the normal modes of small oscillations for this system. For numerical evaluation, you may assume that k=k'.

PHY 711 -- Assignment #13

Oct. 7, 2014

Continue reading Chapter 7 in Fetter & Walecka and the lecture notes.

Consider the eigenvalue problem for u(x) over the interval 0 ≤ x ≤ a such that u(0)=u(a)=0:
d²u_n/dx² = -λ_n u_n(x).
1. Find the smallest eigenvalue λ₁.
2. Use the Rayleigh-Ritz approximation to estimate λ₁ with one or more trial functions (such as u(x)=x(a-x)).

No Title PHY 711 - Assignment #14

10/13/2014

PDF Version

Finish reading Chapter 7 in Fetter and Walecka.

Use the Laplace transform method to solve the following differential equation for ϕ(x) with initial conditions ϕ(0)=0 and d ϕ(0)/dx = 0.

⎛
⎝ − d²
dx²
− 4 ⎞
⎠ ϕ(x) = F₀,

where F₀ is a given constant.

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On 12 Oct 2014, 09:34.

No Title

10/15/2014

PHY 711 - Homework # 15

PDF Version

Assume that a > 0 and use contour integration methods to evaluate the integral:

⌠
⌡ ∞

0
cos(ax)
4x⁴+5x²+1
dx.

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On 12 Oct 2014, 09:46.

PHY 711 -- Assignment #16

Oct. 20, 2014

Complete reading Chap. 7 and Appendix A in Fetter & Walecka.

Note: This version was corrected on 11/07/2014 thanks to Richie Dudley.

Find the Fourier series coefficients for the following discontinuous periodic function (N → ∞):
f(t)=g(t-N2T) + g(t-(N-1)2T) ... g(t-2T)+g(t)+g(t+2T) ... g(t+N2T)
where
g(t) = t/T for t in the interval -T ≤ t ≤ T and 0 otherwise.
Using Maple or its equivalent, plot f(t) and its Fourier representation using the first 3 terms and the first 5 terms of the series.

No Title

Oct 22, 2014

PHY 711 - Problem Set # 17

Start reading Chapter 5 in Fetter and Walecka.

The above figure shows an object with four particles held together with massless bonds at the coordinates shown. The masses of the particles are m₁=m₂ ≡ 2m and m₃=m₄ ≡ m.

Evaluate the moment of inertia tensor for this object in the given coordinate system.
Find the principle moments of inertia and the corresponding principle axes. Sketch the location of the axes.

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On 22 Oct 2014, 12:32.

PHY 711 -- Assignment #18

Oct. 24, 2014

Continue reading Chapter 5 in Fetter & Walecka.

Work Problem 5.9, found at the end of Chapter 5 in your textbook.

PHY 711 -- Assignment #19

Oct. 29, 2014

Finish reading Chapter 8 in Fetter & Walecka.

Work Problem 8.7a found at the end of Chapter 8 in your textbook. Consider the specific case, that the amplitude function has the form ρ(r, φ)= R(r) cos( φ), where the radial function can be expressed in Bessel and Neumann functions. Find the numerical value of the normal mode frequency assuming that a=0.1 m and b=0.2 m, and plot the radial function R(r).

No Title

10/31/2014

PHY 711 - Homework # 20 PDF Version.

Start reading Chap. 9 in Fetter and Walecka.

Consider a velocity field v(r,t). "Derive" the identity

( v ·∇) v = ∇ ⎛
⎝ 1
2
v² ⎞
⎠ − v ×( ∇×v ).

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On 30 Oct 2014, 21:56.

PHY 711 -- Assignment #21

Nov. 03, 2014

Continue reading Chapter 9 in Fetter & Walecka.

Determine the form of the velocity potential for an incompressible fluid representing uniform velocity in the z direction at large distances from a spherical obstruction of radius a. Find the form of the velocity potential and the velocity field for all r > a. Assume that the velocity in the radial direction is 0 for r = a and assume that the velocity is uniform in the azimuthal direction.

No Title

11/17/2014

PDF version PHY 711 - Homework # 22

Start reading Chap. 11 in Fetter and Walecka.

In class, we showed how the equations for a non-linear wave on the surface of an incompressible fluid could be approximated by the famous Korteweg–de Vries equation

⎛
⎝

1−

c²

⎞
⎠

η(u) −

h²

d² η(u)

du²

−

(η(u))² = 0.

Here g is the gravitational acceleration, h is the average height of the water, and c is the constant velocity of the wave to be determined and in terms of position in time u ≡ x − ct.

For an assumed amplitude η₀, show that the following function satisfies the Korteweb-de Vries equation:

η(u) = η₀
cosh² ⎛
⎝
√

[(3 η₀)/h]

u
2h
⎞
⎠
.

The consistent value of the wave speed is given by

c²= gh
1−η₀/h
.
In class, we also showed that to the lowest order approximation, the velocity potential function was related to the wave form function η(u) is given by

d χ(u)
du
= − g
c
η(u).

From this relationship, find χ(u).

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On 15 Nov 2014, 18:44.

PHY 711 -- Assignment #23

Nov. 19, 2014

Start reading Chapter 12 in Fetter & Walecka.

Work Problem 11.1a found at the end of Chapter 11 in your textbook.

PHY 711 -- Assignment #24

Nov. 21, 2014

Continue reading Chapter 12 in Fetter & Walecka.

Work out the intermediate steps for deriving Stokes' law of viscous drag.

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Last modfied: Thursday, 20-Nov-2014 20:58:52 EST