PHY 711 Classical mechanics and mathematical methods

PHY 711 Classical Mechanics and Mathematical Methods

MWF 9 AM-9:50 AM

OPL 107

http://www.wfu.edu/~natalie/f17phy711/

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	Due
1	Mon, 8/28/2017	Chap. 1	Introduction	#1	9/6/2017
2	Wed, 8/30/2017	Chap. 1	Scattering theory	#2	9/6/2017
3	Fri, 9/01/2017	Chap. 1	Scattering theory
4	Mon, 9/04/2017	Chap. 1	Scattering theory	#3	9/6/2017
5	Wed, 9/06/2017	Chap. 2	Physics in an non-inertial reference frame	#4	9/8/2017
6	Fri, 9/08/2017	Chap. 3	Calculus of variations	#5	9/11/2017
7	Mon, 9/11/2017	Chap.3	Calculus of variations	#6	9/13/2017
8	Wed, 9/13/2017	Chap. 3	Lagrangian Mechanics	#7	9/15/2017
9	Fri, 9/15/2017	Chap. 3 and 6	Lagrangian mechanics and constraints	#8	9/20/2017
10	Mon, 9/18/2017	Chap. 3 and 6	Constants of the motion
11	Wed, 9/20/2017	Chap. 3 and 6	Hamiltonian formalism	#9	9/27/2017
12	Fri, 9/22/2017	Chap. 3 and 6	Liouville equation
13	Mon, 9/25/2017	Chap. 6	Canonical transformations
14	Wed, 9/27/2017	Chap. 4	Motion from Small oscillations about equilibrium
15	Fri, 9/29/2017	Chap. 1-4,6	Review
	Mon, 10/02/2017		Take-home exam -- No class
	Wed, 10/04/2017		Take-home exam -- No class
16	Fri, 10/06/2017	Chap. 4	Vibrational modes
17	Mon, 10/09/2017	Chap. 5	Rigid body motion	#10	10/16/2017

18	Wed, 10/11/2017	Chap. 5	Rigid body motion
	Fri, 10/13/2017		Fall break -- No class
19	Mon, 10/16/2017		Discuss exam questions and topics for presentations	Topic	10/18/2017
20	Wed, 10/18/2017	Chap. 7	Wave equation in one dimension	#11	10/20/2017
21	Fri, 10/20/2017	Chap. 7	Solutions of Sturm-Liouville equations	#12	10/27/2017
22	Mon, 10/23/2017	Chap. 7	Solutions of Sturm-Liouville equations
23	Wed, 10/25/2017	Chap. 7	Solutions of Sturm-Liouville equations
24	Fri, 10/27/2017	App. A	Laplace transforms and contour integrals	#13	11/01/2017
25	Mon, 10/30/2017	App. A	Contour integrals
26	Wed, 11/01/2017	Chap. 8	Mechanics of Elastic Membranes	#14	11/06/2017
27	Fri, 11/03/2017	Chap. 9	Introduction to hydrodynamics
28	Mon, 11/06/2017	Chap. 9	Introduction to hydrodynamics	#15	11/10/2017
29	Wed, 11/08/2017	Chap. 9	Sound waves
30	Fri, 11/10/2017	Chap. 9	Sound waves	#16	11/17/2017
	Mon, 11/13/2017		Class cancelled
31	Wed, 11/15/2017	Chap. 9	Sound waves -- including non-linearities
32	Fri, 11/17/2017	Chap. 10	Surface waves in fluids	#17	11/27/2017
33	Mon, 11/20/2017	Chap. 10	Surface waves in fluids
	Wed, 11/22/2017		Thanksgiving Holiday -- No class
	Fri, 11/24/2017		Thanksgiving Holiday -- No class
34	Mon, 11/27/2017	Chap. 11	Heat conductivity
35	Wed, 11/29/2017	Chap. 12	Viscous fluids
36	Fri, 12/01/2017	Chap. 12	Viscous fluids
	Mon, 12/04/2017		Presentations I
	Wed, 12/06/2017		Presentations II
	Fri, 12/08/2017		Presentations III

No Title PHY 711 - Assignment #1 PDF VERSION

08/28/2017

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 27 Aug 2017, 10:53.

PHY 711 -- Assignment #2

Aug. 30, 2017

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. 4, 2017

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. 6, 2017

Read Chapter 2 in Fetter & Walecka.

Suppose that you would like to install a Foucault Pendulum at a location of your choice. Find the latitude of your location and determine the period of the pendulum to make a complete circle in the direction of its swing.

PHY 711 -- Assignment #5

Sept. 8, 2017

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=2, y=7).

No Title PHY 711 - Assignment #6 PDF VERSION

Sep 9, 2017

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 9 Sep 2017, 18:14.

No Title PHY 711 - Assignment #7 PDF VERSION

Sep 12, 2017

Consider a Lagrangian describing the motion of a particle of mass m and charge q given by

L(x,y,z,
⋅

x

,
⋅

y

,
⋅

z

) = 1
2
m ⎛
⎝
⋅

x

2

+
⋅

y

2

+
⋅

z

2

⎞
⎠ + q
c
B
⋅

y

x.

Here c denotes the speed of light and B represents the magnitude of a constant magnetic field along the z-axis. Determine the Euler-Lagrange equations of motion for the particle and discuss how the motion compares with the similar example discussed in class.

File translated from T_EX by T_TH, version 4.01.
On 12 Sep 2017, 22:30.

No Title PHY 711 - Assignment #8 PDF VERSION

9/15/2017

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 14 Sep 2017, 20:33.

PHY 711 -- Assignment #9

Sept. 20, 2017

Continue reading Chapters 3 and 6 in Fetter & Walecka.

Work problem 6.5 at the end of Chapter 6 of F & W.

PHY 711 -- Assignment #10

Oct. 9, 2017

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 principal moments of inertia and the corresponding principal axes. Sketch the location of the axes.

No Title PHY 711 - Assignment #11

PDF VERSION

10/18/2017

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 17 Oct 2017, 17:15.

PHY 711 -- Assignment #12

Oct. 20,, 2017

Continue reading Chapter 7 in Fetter & Walecka.

Work problem 7.1 at the end of Chap. 7 in Fetter & Walecka.

No Title

10/27/2017

PDF VERSION PHY 711 - Homework # 13

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.

File translated from T_EX by T_TH, version 4.01.
On 26 Oct 2017, 21:42.

PHY 711 -- Assignment #14

Nov. 01, 2017

Read Chapter 8 in Fetter & Walecka.

Work problem 8.5 at the end of Chap. 8 in Fetter & Walecka

PHY 711 -- Assignment #15

Nov. 6, 2017

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 #16

Nov. 10, 2017

Continue reading Chapter 9 in Fetter & Walecka.

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

PHY 711 -- Assignment #17

Nov. 17, 2017

Start reading Chapter 10 in Fetter & Walecka.

Work Problem 10.3 at the end of Chapter 10 in Fetter and Walecka. If helpful, you may wish to use some of the material presented in Lecture 32.

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Last modfied: Thursday, 16-Nov-2017 20:39:06 EST