PHY 711 Classical mechanics and mathematical methods

PHY 711 Classical Mechanics and Mathematical Methods

MWF 11 AM-11:50 AM

OPL 107

http://www.wfu.edu/~natalie/f16phy711/

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

Course schedule

(Preliminary schedule -- subject to frequent adjustment.) i

	Date	F&W Reading	Topic	Assignment	Due
1	Wed, 8/31/2016	Chap. 1	Review of basic principles	#1	9/7/2016
2	Fri, 9/02/2016	Chap. 1	Scattering theory	#2	9/7/2016
	Mon, 9/05/2016		Labor day -- no class
3	Wed, 9/07/2016	Chap. 1	Scattering theory	#3	9/9/2016
4	Fri, 9/09/2016	Chap. 1 & 2	Scattering theory and rotations	#4	9/12/2016
5	Mon, 9/12/2016	Chap. 3	Calculus of variations	#5	9/14/2016
6	Wed, 9/14/2016	Chap. 3	Calculus of variations	#6	9/16/2016
7	Fri, 9/16/2016	Chap. 3	Lagrangian mechanics	#7	9/19/2016
8	Mon, 9/19/2016	Chap. 3 and 6	Lagrangian mechanics and constraints	#8	9/21/2016
9	Wed, 9/21/2016	Chap. 3 and 6	Constants of the motion	#9	9/23/2016
10	Fri, 9/23/2016	Chap. 3 and 6	Hamiltonian and canonical equations of motion	#10	9/26/2016
11	Mon, 9/26/2016	Chap. 3 and 6	Phase space	#11	9/28/2016
12	Wed, 9/28/2016	Chap. 6	Canonical transformations	#12	9/30/2016
13	Fri, 9/30/2016	Chap. 4	Small oscillations	#13	10/04/2016
14	Tue, 10/04/2016	Chap. 4	Normal modes	#14	10/07/2016
15	Wed, 10/05/2016	Chap. 7	Wave motion in one dimension	#15	10/07/2016
16	Fri, 10/07/2016	Chap. 7	Sturm-Liouville equations
17	Mon, 10/10/2016	Chap. 7	Sturm-Liouville equations	Take-home exam
18	Wed, 10/12/2016	Chap. 7	Fourier series and transforms	Take-home exam
19	Fri, 10/14/2016	App. A	Laplace transforms and contour integrals	Take-home exam
20	Mon, 10/17/2016	Chap. 5	Mechanics of rigid bodies	Exam due
21	Wed, 10/19/2016	Chap. 5	Mechanics of rigid bodies	#16	10/24/2016
	Fri, 10/21/2016		Fall break -- no class
22	Mon, 10/24/2016	Chap. 8	Mechanics of Elastic Membranes	#17	10/28/2016
23	Wed, 10/26/2016	Chap. 9	Introduction to hydrodynamics
24	Fri, 10/28/2016	Chap. 9	Introduction to hydrodynamics	#18	10/31/2016
25	Mon, 10/31/2016	Chap. 9	Sound waves	#19	11/02/2016
26	Wed, 11/02/2016	Chap. 9	Sound waves	#20	11/04/2016
27	Fri, 11/04/2016	Chap. 9	Non-linear sound	#21	11/07/2016
28	Mon, 11/07/2016	Chap. 10	Surface waves in fluids
29	Wed, 11/09/2016	Chap. 10	Surface waves in fluids	#22	11/11/2016
30	Fri, 11/11/2016	Chap. 11	Heat conductivity	#23	11/14/2016
31	Mon, 11/14/2016	Chap. 12	Viscous fluids	#24	11/16/2016
32	Wed, 11/16/2016	Chap. 12	Viscous fluids	#25	11/18/2016
33	Fri, 11/18/2016	Chap. 12	Viscous fluids	#26	11/21/2016
34	Mon, 11/21/2016	Chap. 13	Elastic continua	Prepare presentations
	Wed, 11/23/2016		Thanksgiving Holiday -- no class
	Fri, 11/25/2016		Thanksgiving Holiday -- no class

35	Mon, 11/28/2016	Chap. 13	Elastic continua	Prepare presentations
36	Wed, 11/30/2016		Math methods	Prepare presentations
37	Fri, 12/02/2016		Math methods	Prepare presentations
38	Mon, 12/05/2016		Review	Prepare presentations
	Wed, 12/07/2016		Presentations I
	Fri, 12/09/2016		Presentations II

No Title PHY 711 - Assignment #1 PDF VERSION

08/31/2016

Use maple or mathematica to plot the functions

f(x)=e^−x² and h(x)= ⌠
⌡ x

0
f(t) dt.

and to numerically evaluate f(5) and h(5).

File translated from T_EX by T_TH, version 4.01.
On 31 Aug 2016, 09:05.

PHY 711 -- Assignment #2

Sept. 2, 2016

Read Chapter 1 in Fetter & Walecka.

In class, we "derived" the differential cross section for the scattering of two hard spheres of mutual radius D in the center of mass frame. Find the differential cross section for this system in the lab frame in which m_target is initially at rest and evaluate the expression for the following cases.
- m₁/m_target=0.1
- m₁/m_target=1
- m₁/m_target=1000

PHY 711 -- Assignment #3

Sept. 7, 2016

Continue reading Chapter 1 in Fetter & Walecka.

Work Problem #1.15 at the end of Chapter 1 in Fetter and Walecka.

PHY 711 -- Assignment #4

Sept. 9, 2016

Finish reading Chapter 1 and start reading Chapter 2 in Fetter & Walecka.

Work Problem #1.16 at the end of Chapter 1 in Fetter and Walecka.

PHY 711 -- Assignment #5

Sept. 12, 2016

Start reading Chapter 3, especially Section 17, in Fetter & Walecka.

Using calculus of variations, find the equation y(x) of the shortest "curve" which passes through the points (x=0, y=0) and (x=3, y=7).

No Title PHY 711 - Assignment #6 PDF VERSION

Sep 13, 2016

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
  
  2
  
  t², 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.

File translated from T_EX by T_TH, version 4.01.
On 13 Sep 2016, 18:13.

PHY 711 -- Assignment #7

Sept. 16, 2016

Continue reading Chapter 3 in Fetter & Walecka.

Work Problem #3.3 at the end of Chapter 3 in Fetter and Walecka. It is not necessary to solve the full equations of motion beyond estimating the frequency of small oscillations of the pendulum.

No Title PHY 711 - Assignment #8

9/19/2016

PDF VERSION

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 the fixed constants of the system (θ, m, M, etc.) 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.)

File translated from T_EX by T_TH, version 4.01.
On 18 Sep 2016, 20:52.

No Title PHY 711 - Assignment #9

9/21/2016

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. Extra credit: Find an equivalent Lagrangian in terms of one generalized coordinate and its velocity and the constants of motion for the system.

File translated from T_EX by T_TH, version 4.01.
On 20 Sep 2016, 20:22.

PHY 711 -- Assignment #10

Sept. 23, 2016

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 #11

Sept. 26, 2016

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

Read the beginning part of S. Nosé's 1984 paper "A molecular dynamics method for simulations in the canonical ensemble". Work out the details of the equations 2.4-2.10 of the paper.

PHY 711 -- Assignment #12

Sept. 28, 2016

Complete the reading of Chapter 6 in Fetter & Walecka.

Work problem 6.11 at the end of Chap. 6.

PHY 711 -- Assignment #13

Sept. 30, 2016

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 #14

Oct. 4, 2016

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 #15

10/5/2016

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.

File translated from T_EX by T_TH, version 4.01.
On 3 Oct 2016, 21:25.

PHY 711 -- Assignment #16

Oct. 19, 2016

Start Chapter 5 in Fetter & 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.

PHY 711 -- Assignment #17

Oct. 24, 2016

Read Chap. 8 in Fetter & Walecka.

Work problem 8.5 at the end of Chapter 8 in Fetter and Walecka.

PHY 711 -- Assignment #18

Oct. 28, 2018

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 #19

Oct. 31, 2016

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 #20

Nov. 02, 2016

Continue reading Chapter 9 in Fetter & Walecka.

Work Problem 9.2 at the end of Chapter 9 in Fetter and Walecka.

PHY 711 -- Assignment #21

Nov. 04, 2016

Continue reading Chapter 9 in Fetter & Walecka.

Consider a one-dimensional nonlinear sound wave in the adiabatic approximation as discussed in Lecture 27. Using Maple or other software, visualize the time evolution of a waveform of your choice. You may wish to use the Maple script nonlinearsound.mw or the PDF file as a starting point.

PHY 711 -- Assignment #22

November 9, 2016

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 evaluated at the surface evaluated to the same level of approximation.

PHY 711 -- Assignment #23

November 11, 2016

Start reading Chapter 11 in Fetter & Walecka.

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

PHY 711 -- Assignment #24

November 14, 2016

Start reading Chapter 12 in Fetter & Walecka.

Work problem 12.4 at the end of Chapter 12 in Fetter & Walecka.

PHY 711 -- Assignment #25

November 16, 2016

Start reading Chapter 12 in Fetter & Walecka.

Following the lecturenotes or other sources, derive Stokes law for viscous drag.

PHY 711 -- Assignment #26

November 16, 2016

Continue reading Chapter 12 in Fetter & Walecka.

Following the lecturenotes and your textbook, find the form of the entropy wave corresponding to the longitudinal mode of sound in the presence of visosity corresponding to Eq. 62.21 of your text.

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Last modfied: Sunday, 04-Dec-2016 17:40:18 EST