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/f15phy711/

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

Course schedule

(Preliminary schedule -- subject to frequent adjustment.)

	Date	F&W Reading	Topic	Assignment
1	Wed, 8/26/2015	Chap. 1	Review of basic principles	#1
2	Fri, 8/28/2015	Chap. 1	Scattering theory	#2
3	Mon, 8/31/2015	Chap. 1	Scattering theory continued	#3
4	Wed, 9/02/2015	Chap. 2	Accelerated coordinate systems	#4
5	Fri, 9/04/2015	Chap. 3	Calculus of variations	#5
6	Mon, 9/07/2015	Chap. 3	Calculus of variations	#6
7	Wed, 9/09/2015	Chap. 3	Hamilton's principle	#7
8	Fri, 9/11/2015	Chap. 3 & 6	Hamilton's principle	#8
9	Mon, 9/14/2015	Chap. 3 & 6	Lagrangians with constraints	#9
10	Wed, 9/16/2015	Chap. 3 & 6	Lagrangians and constants of motion	#10
11	Fri, 9/18/2015	Chap. 3 & 6	Hamiltonian formalism	#11
12	Mon, 9/21/2015	Chap. 3 & 6	Hamiltonian formalism	#12
13	Wed, 9/23/2015	Chap. 3 & 6	Hamiltonian Jacobi transformations	#13
14	Fri, 9/25/2015	Chap. 4	Small oscillations	#14
15	Mon, 9/28/2015	Chap. 4	Normal modes of motion	#15
16	Wed, 9/30/2015	Chap. 7	Wave motion	#16
17	Fri, 10/02/2015	Chap. 7 & App. A	Contour Integration	#17
18	Mon, 10/05/2015	Chap. 7	Fourier transforms	#18
19	Wed, 10/07/2015	Chap. 7	Laplace transforms	#19
20	Fri, 10/09/2015	Chap. 7	Green's functions	Start exam
	Mon, 10/12/2015		No class	Take home exam
	Wed, 10/14/2015		No class	Exam due before 10/19/2015
	Fri, 10/16/2015		Fall break -- no class
21	Mon, 10/19/2015	Chap. 5	Motion of Rigid Bodies	#20
22	Wed, 10/21/2015	Chap. 5	Motion of Rigid Bodies	#21
23	Fri, 10/23/2015	Chap. 8	Motion of Elastic membranes	#22
24	Mon, 10/26/2015	Chap. 9	Hydrodynamics	#23
25	Wed, 10/28/2015	Chap. 9	Hydrodynamics	#24
26	Fri, 10/30/2015	Chap. 9	Sound waves	#25
27	Mon, 11/02/2015	Chap. 9	Sound waves	#26
28	Wed, 11/04/2015	Chap. 9	Sound waves
29	Fri, 11/06/2015	Chap. 10	Surface waves on fluids	#27
30	Mon, 11/09/2015	Chap. 10	Surface waves on fluids	#28
31	Wed, 11/11/2015	Chap. 11	Heat Conduction	#29
32	Fri, 11/13/2015	Chap. 12	Viscosity	#30
33	Mon, 11/16/2015	Chap. 12	Viscosity	Prepare presentation.
34	Wed, 11/18/2015	Chap. 12	Viscosity	Prepare presentation.
35	Fri, 11/20/2015	Chap. 12	Elastic Continua	Prepare presentation.
36	Mon, 11/23/2015	Chap. 13	Review of Mathematical Methods	Prepare presentation.
	Wed, 11/25/2015		Thanksgiving Holiday
	Fri, 11/27/2015		Thanksgiving Holiday
37	Mon, 11/30/2015	Chap. 13	Review of Mathematical Methods	Prepare presentation.

	Wed, 12/02/2015		Student presentations I
	Fri, 12/04/2015		Student presentations II
	Mon, 12/07/2015		Begin Take-home final

No Title PHY 711 - Assignment #1

08/26/2015

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 25 Aug 2015, 21:47.

No Title PHY 711 - Assignment #2

PDF VERSION

8/28/2015

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 27 Aug 2015, 21:43.

No Title PHY 711 - Assignment #3

08/31/2015

PDF VERSION

In class we evaluated the differential cross section for Rutherford scattering. A slightly different evaluation uses the integral

θ = π−2b

⌠
⌡

∞

r_min

1/r²

⎛
√

1 −

b²

r²

−

= π− 2b

⌠
⌡

∞

r_min

1/r

√

r² −b²−κr

Here r_min is a solution of the equation

1 −

b²

r²_min

−

r_min

=0.

Here b is the impact parameter which is a function of the scattering angle θ. κ is a length parameter with represents the ration of the interaction strength to the center of mass energy of target and scattering particles. Use Maple or other algebraic manipulation software to evaluate the integral to show that

2b=

tan(θ/2)

From this form of the impact parameter b(θ), "derive" the Rutherford scattering cross section.

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On 30 Aug 2015, 00:37.

PHY 711 -- Assignment #4

Sept. 2, 2015

Read Chapter 2 in Fetter & Walecka.

Suppose you wanted to install a Foucault Pendulum on campus. Estimate the pendulum period for Wake Forest University which is located at lattitude 36 deg and longitude -80 deg.

PHY 711 -- Assignment #5

Sept. 4, 2015

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 3, 2015

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 the expression for f as an explicit function of t ( f(t) ) and take its time derivative directly to check your previous results.

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On 3 Sep 2015, 01:25.

PHY 711 -- Assignment #7

Sept. 9, 2015

Continue reading Chapter 3 in Fetter & Walecka.

Work Problem #3.13 at the end of Chap. 3 in Fetter & Walecka.

PHY 711 -- Assignment #8

Sept. 11, 2015

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 PDF VERSION

9/14/2015

Continue reading Chapters 3 and 6 in Fetter and Walecka.

The figure above shows a box of mass m 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. (Note that X and s represent components of vectors whose directions are related by the angle θ.)

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On 13 Sep 2015, 15:20.

No Title PHY 711 - Assignment #10

9/16/2015

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 15 Sep 2015, 17:53.

PHY 711 -- Assignment #11

Sept. 18, 2015

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

Sept. 21, 2015

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

Follow the paper by H. C. Andersen and "derive" Eqs. 3.14.

PHY 711 -- Assignment #13

Sept. 23, 2015

Finish reading Chapter 6 in Fetter & Walecka.

Verify the Hamilton-Jacobi solution to the harmonic oscillator problem we covered in class.
Simplify the expression for the action S(q,t) to show that it is consistent with the action calculated directly from the Lagrangian.

PHY 711 -- Assignment #14

Sept. 25, 2015

Start reading Chapter 4 in Fetter & Walecka.

Consider the the mass and spring system described by Eq. 24.1 and Fig. 24.1 in Fetter & Walecka. Explicitly consider the cases of N=3 and N=4. Compare the normal mode eigenvalues for these two cases (obtained with the help of Maple or Mathematica) with the equivalent analysis given by Eq. 24.38.

PHY 711 -- Assignment #15

Sept. 28, 2015

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'.

No Title PHY 711 - Assignment #16

9/30/2015

PDF VERSION

Continue reading Chapter 7 in Fetter and Walecka.

Consider a displacement function u(x,t) representing a one-dimensional traveling wave (either transverse or longitudinal) which is a solution of the one-dimensional wave equation with wave speed c:

∂² u
∂x²
− 1
c²
∂² u
∂t²
=0.

If the initial conditions for the wave displacement u(x,t) are given by

u(x,0) = U₀ e^{−(x−x₀)²/σ²},

and

∂u
∂t
(x,0) = V₀ ⎛
⎝ x
μ
⎞
⎠ 3

e^−(x/μ)⁴,

find the form of u(x,t) for t > 0. Express your result in terms of the constants U₀, V₀, σ, μ, x₀, and c.

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On 29 Sep 2015, 16:12.

No Title

10/02/2015

PHY 711 - Homework # 17

PDF VERSION Read Appendix A of Fetter and Walecka.

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 1 Oct 2015, 21:27.

PHY 711 -- Assignment #18

Oct. 5, 2015

Continue reading Chap. 7 in Fetter & Walecka.

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 PHY 711 - Assignment #19

PDF VERSION

10/7/2015

Continue 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 4 Oct 2015, 00:23.

No Title

Oct 19, 2015

PHY 711 - Problem Set # 20

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.

No Title

Oct 20, 2015

PHY 711 - Problem Set # 21 PDF VERSION

Finish reading Chapter 5 in Fetter and Walecka.

In most Classsical Mechanics texts (besides Fetter and Walecka), the Euler angles are defined with a different convention as shown below. (This figure was slightly modified from one available on the website http://en.wikipedia.org/wiki/Euler_angles.)

In this case, the first rotation is about the original ∧z axis by ϕ corresponding to the rotation matrix

ℜ_ϕ =

⎛
⎜
⎜
⎜
⎝

cosϕ

sinϕ

−sinϕ

cosϕ

⎞
⎟
⎟
⎟
⎠

(1)

The second rotation is about the new ∧x axis by θ corresponding to the rotation matrix

ℜ_θ =

⎛
⎜
⎜
⎜
⎝

cosθ

sinθ

−sinθ

cosθ

⎞
⎟
⎟
⎟
⎠

(2)

In this case, the last rotation is about the new ∧z axis by ψ corresponding to the rotation matrix

ℜ_ψ =

⎛
⎜
⎜
⎜
⎝

cosψ

sinψ

−sinψ

cosψ

⎞
⎟
⎟
⎟
⎠

(3)

For this convention, write a general expression for the angular velocity vector ω in terms of the time rate of change of these Euler angles - · ϕ, · θ, and · ψ corresponding to the 29.7 of your text.

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On 20 Oct 2015, 22:30.

PHY 711 -- Assignment #22

Oct. 23, 2015

Read Chap. 8 in Fetter & Walecka.

Work problem 8.2 at the end of Chapter 8 in Fetter and Walecka.

No Title

10/26/2015

PDF VERSION PHY 711 - Homework # 23

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 ).

File translated from T_EX by T_TH, version 4.01.
On 25 Oct 2015, 20:19.

PHY 711 -- Assignment #24

Oct. 28, 2015

Continue reading Chapter 9 in Fetter & Walecka.

Determine the form of the velocity potential for a 3-dimensional compressible fluid which flows at 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.

PHY 711 -- Assignment #25

Oct. 30, 2015

Continue reading Chapter 9 in Fetter & Walecka.

Consider the analysis for the speed of sound in an ideal gas as we derived in class. Using the subscript "0" to denote "stardard" conditions with temperature T₀=273.15 K and pressure p₀= 10⁵ Pa, evaluate c₀ and plot c/c₀ as a function of ρ/ρ₀ for
1. He ( γ=1.6666666)
2. O₂ ( γ=1.4)

PHY 711 -- Assignment #26

Nov. 2, 2015

Continue reading Chapter 9 in Fetter & Walecka.

Consider the treatment of the scattering of a sound plane wave by a cylindrical obstacle in the section "Scattering by a Rigid Cylinder" of Chapter 9. For the case that ka << 1, the scattering amplitude and cross sections reduce to Eq. 51.76 and 51.77. Add to these expressions the contributions from the |m|=2 terms.

PHY 711 -- Assignment #27

November 6, 2015

Start reading Chapter 10 in Fetter & Walecka.

Work problem 10.3 in Fetter & Walecka. (Hint: Think about how you might be able to apply a version of Eq. 53.15 for this case.)

PHY 711 -- Assignment #28

November 9, 2015

Continue reading Chapter 10 in Fetter & Walecka.

Consider the soliton solution of the non-linear surface wave equations where the vertical displacement waveform is given by Eq. 56.40 in Fetter & Walecka. Find the corresponding velocity potential to the same level of approximation.

PHY 711 -- Assignment #29

November 11, 2015

Start reading Chapter 11 in Fetter & Walecka.

Work problem 11.1 at the end of Chapter 11 in Fetter & Walecka.

PHY 711 -- Assignment #30

November 13, 2015

Start reading Chapter 12 in Fetter & Walecka.

Work problem 12.3a at the end of Chapter 12 in Fetter & Walecka.

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Last modfied: Friday, 20-Nov-2015 00:52:12 EST