PHY 711 Classical mechanics and mathematical methods

PHY 711 Classical Mechanics and Mathematical Methods

MWF 10 AM-10:50 AM

OPL 103

http://www.wfu.edu/~natalie/f18phy711/

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, 8/27/2018	Chap. 1	Introduction	#1	9/7/2018
	Wed, 8/29/2018	No class
2	Fri, 8/31/2018	Chap. 1	Scattering theory	#2	9/7/2018
3	Mon, 9/03/2018	Chap. 1	Scattering theory
4	Wed, 9/05/2018	Chap. 1	Scattering theory	#3	9/10/2018
5	Fri, 9/07/2018	Chap. 2	Non-inertial coordinate systems	#4	9/12/2018
6	Mon, 9/10/2018	Chap. 3	Calculus of Variation	#5	9/12/2018
7	Wed, 9/12/2018	Chap. 3	Calculus of Variation	#6	9/17/2018
	Fri, 9/14/2018	No class	University closed due to weather.
8	Mon, 9/17/2018	Chap. 3	Lagrangian Mechanics	#7	9/19/2018
9	Wed, 9/19/2018	Chap. 3 and 6	Lagrangian Mechanics and constraints	#8	9/24/2018
10	Fri, 9/21/2018	Chap. 3 and 6	Constants of the motion
11	Mon, 9/24/2018	Chap. 3 and 6	Hamiltonian formalism	#9	9/28/2018
12	Wed, 9/26/2018	Chap. 3 and 6	Liouville theorem	#10	10/3/2018
13	Fri, 9/28/2018	Chap. 3 and 6	Canonical transformations
14	Mon, 10/1/2018	Chap. 4	Small oscillations about equilibrium	#11	10/5/2018
15	Wed, 10/3/2018	Chap. 4	Normal modes of vibration
16	Fri, 10/5/2018	Chap. 1-4, 6	Review
17	Mon, 10/8/2018	Chap. 7	Strings
18	Wed, 10/10/2018	Chap. 7	Wave equation
	Fri, 10/12/2018	No class	Fall break
19	Mon, 10/15/2018	Chap. 7	Wave equation
20	Wed, 10/17/2018	Chap. 7	Fourier Transforms	#12	10/22/2018
21	Fri, 10/19/2018	Chap. 7	Laplace transforms; Contour integrals	#13	10/24/2018
22	Mon, 10/22/2018	Chap. 7	Contour integrals
23	Wed, 10/24/2018	Chap. 5	Rigid body motion	#14	10/26/2018
24	Fri, 10/26/2018	Chap. 5	Rigid body motion	#15	10/31/2018
25	Mon, 10/29/2018	Chap. 8	Mechanics of elastic membranes	#16	11/02/2018
26	Wed, 10/31/2018	Chap. 9	Mechanics of three dimensional fluids
27	Fri, 11/02/2018	Chap. 9	Mechanics of fluids	#17	11/07/2018
28	Mon, 11/05/2018	Chap. 9	Sound waves	Project topic
29	Wed, 11/07/2018	Chap. 9	Sound waves	#18	11/12/2018
30	Fri, 11/09/2018	Chap. 9	Linear and non-linear sound
31	Mon, 11/12/2018	Chap. 10	Surface waves	#19	11/16/2018
32	Wed, 11/14/2018	Chap. 10	Surface waves -- nonlinear effects
33	Fri, 11/16/2018	Chap. 11	Heat conductivity	#20	11/26/2018
34	Mon, 11/19/2018	Chap. 13	Elastic media
	Wed, 11/21/2018	No class	Thanksgiving holiday
	Fri, 11/23/2018	No class	Thanksgiving holiday
35	Mon, 11/26/2018	Chap. 12	Viscous fluids
36	Wed, 11/28/2018	Chap. 12	Viscous fluids
37	Fri, 11/30/2018		Review
	Mon, 12/03/2018		Presentations I
	Wed, 12/05/2018		Presentations II
	Fri, 12/07/2018		Presentations III

PHY 711 – Assignment #1

08/27/2018

Use maple or mathematica to plot the functions $∫ -x2 x f(x) = e and h(x) = 0 f (t) dt.$
and to numerically evaluate f(5) and h(5).

PHY 711 -- Assignment #2

Aug. 31, 2018

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. Analyze this system directly in the laboratory frame in which the target of mass m_target is initially at rest and the scattering particle has mass m and initial velocity V₀. Consider the following two cases, finding the differential cross section as a function of lab angle for each (You may wish to check your answers using the center of mass expressions.)
- m << m_target
- m = m_target

PHY 711 -- Assignment #3

Sept. 5, 2018

Continue reading Chapter 1 in Fetter & Walecka.

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

PHY 711 -- Assignment #4

Sept. 7, 2018

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. 10, 2018

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=4).

PHY 711 – Assignment #6

September 11, 2018

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

Consider an arbitrary function of the form f = f(q,,t), where it is assumed that q = q(t) and ≡ dq∕dt.
1. Evaluate $∂ df d ∂f ------ -----. ∂q dt dt∂q$
2. Evaluate $∂-df-- d-∂f-. ∂ ˙q dt dt∂ ˙q$
3. Evaluate $df ---. dt$
4. Now suppose that $2 2 -t∕τ f (q,q˙, t) = qq˙t , where q(t) = e .$
  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 and take its time derivative directly to check your previous results.

PHY 711 – Assignment #7

September 16, 2018

Consider a Lagrangian describing the motion of a particle of mass m and charge q given by $1 ( ) q L(x,y, z, ˙x, ˙y,z˙) =-m x˙2 + y˙2 + ˙z2 + --By˙x. 2 c$
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.

PHY 711 – Assignment #8

9/19/2019

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.)

PHY 711 -- Assignment #9

Sept. 24, 2018

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

Sept. 25, 2018

Continue reading Chapters 3 and 6 in Fetter & Walecka.

Choose one of the papers distributed in class, by H. C. Andersen or by S. Nose' and derive to your satisfaction the Euler-Lagrange equations of motion, the Hamiltonian, and the canonical equations of motion for the constant pressure or constant temperature simulations, respectively.

PHY 711 -- Assignment #11

Oct. 1, 2018

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 case of N=4 and find the 4 coupled equations of motion. Compare the normal mode eigenvalues for this case (obtained with the help of Maple or Mathematica) with the equivalent analysis given by Eq. 24.38.

PHY 711 – Assignment #12

10/17/2018

Continue reading Chapter 7 in Fetter and Walecka.

A function which is periodic in the length L is given by $∞ f (x) ≡ √-1-- ∑ exp(- (x + nL)2∕a2), πa n=- ∞$
where a is a real constant.
1. Find the Fourier transform F(q), of f(x).
2. Find a convenient notation to enumerate the discrete values of q.
3. Write an expression to evaluate the largest value of q needed such that the F(q)∕F(0) ≤ 0.01.

10/19/2018

PHY 711 – Homework # 13

Read Appendix A of Fetter and Walecka.

Assume that a > 0 and b > 0; use contour integration methods to evaluate the integral: $∫ ∞ cos(ax) --4----4dx. 0 x + 4b$

PHY 711 -- Assignment #14

Oct. 24, 2018

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.

PHY 711 -- Assignment #15

Oct. 26, 2018

Continue reading Chapter 5 in Fetter & Walecka.

Work problem 5.9 at the end of Chap. 5 of Fetter & Walecka.

PHY 711 -- Assignment #16

Oct. 29, 2018

Read Chapter 8 in Fetter & Walecka.

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

PHY 711 -- Assignment #17

Nov. 2, 2018

Continue reading Chapter 9 in Fetter & Walecka.

Determine the form of the velocity potential for a 3-dimensional incompressible 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 #18

Nov. 7, 2018

Continue reading Chapter 9 in Fetter & Walecka.

Consider the approximate result given in Eq. 51.76 of your text book. Show how the result follows from evaluation of Eq. 51.74 when ka << 1.

PHY 711 -- Assignment #19

Nov. 12, 2018

Start reading Chapter 10 in Fetter & Walecka.

Work Problem 10.3 at the end of Chapter 10 in Fetter and Walecka.

PHY 711 -- Assignment #20

Nov. 16, 2018

Start reading Chapter 11 in Fetter & Walecka.

Consider the rectangular heat conduction problem discussed in Lecture 33, starting with slide 9. Suppose that at time t=0, the temperature profile is
T(x,y,z,0) = T₀(1 + sin(πx/a)).
At what later time t does the rectangle achieve a uniform temperature T₀ with a 10 % fluctuation when the rectangle is made of
1. copper
2. air
You may use the values of thermal diffusivity quoted in class. Assume that a = 0.1 m.

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Last modfied: Saturday, 17-Nov-2018 17:38:00 EST