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/f13phy711/

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, 8/28/2013	Chap. 1	Review of basic principles;Scattering theory	#1
2	Fri, 8/30/2013	Chap. 1	Scattering theory continued	#2
3	Mon, 9/02/2013	Chap. 1	Scattering theory continued	#3
4	Wed, 9/04/2013	Chap. 2	Accelerated Coordinate Systems	#4
5	Fri, 9/06/2013	Chap. 3	Calculus of variations	#5
6	Mon, 9/09/2013	Chap. 3	Calculus of variations -- continued
7	Wed, 9/11/2013	Chap. 3	Calculus of variations applied to Lagrangians	#6
8	Fri, 9/13/2013	Chap. 3	Lagrangian mechanics	#7
9	Mon, 9/16/2013	Chap. 3 & 6	Lagrangian mechanics	#8
10	Wed, 9/18/2013	Chap. 3 & 6	Lagrangian mechanics	#9
11	Fri, 9/20/2013	Chap. 3 & 6	Lagrangian & Hamiltonian mechanics	#10
12	Mon, 9/23/2013	Chap. 3 & 6	Hamiltonian formalism	#11
13	Wed, 9/25/2013	Chap. 3 & 6	Hamiltonian formalism	#12
14	Fri, 9/27/2013	Chap. 3 & 6	Hamiltonian formalism	#13
15	Mon, 9/30/2013	Chap. 4	Small Oscillations	#14
16	Wed, 10/02/2013	Chap. 4	Small Oscillations
17	Fri, 10/04/2013	Chap. 4	Small Oscillations	#15
18	Mon, 10/07/2013	Chap. 4 & 7	Small Oscillations and waves	#16
19	Wed, 10/09/2013	Chap. 7	Wave equation
	Fri, 10/11/2013		No class (Fall Break)
20	Mon, 10/14/2013	Chap. 7	Wave equation (Presentation topic due)	#17
21	Wed, 10/16/2013	Chap. 7	Mathematical methods	#18
22	Fri, 10/18/2013	Chap. 7	Mathematical methods	#19
23	Mon, 10/21/2013	Chap. 5	Rigid rotations	#20
24	Wed, 10/23/2013	Chap. 5	Rigid rotations	#21
25	Fri, 10/25/2013	Chap. 5	Rigid rotations
	Mon, 10/28/2013	No class	Take-home exam
	Wed, 10/30/2013	No class	Take-home exam
	Fri, 11/01/2013	No class	Take-home exam
26	Mon, 11/04/2013	Chap. 8	Oscillations in two-dimensional membranes	Take-home exam due
27	Wed, 11/06/2013	Chap. 9	Physics of fluids	#22
28	Fri, 11/08/2013	Chap. 9	Physics of fluids	#23
29	Mon, 11/11/2013	Chap. 9	Sound Waves	#24
30	Wed, 11/13/2013	Chap. 9	Sound Waves	#25
31	Fri, 11/15/2013	Chap. 9	Non linear effects in Sound	#26
32	Mon, 11/18/2013	Chap. 10	Surface waves
33	Wed, 11/20/2013	Chap. 10	Surface waves
34	Fri, 11/22/2013	Chap. 11	Heat conduction
35	Mon, 11/25/2013	Chap. 12	Viscous fluids
	Wed, 11/27/2013		Thanksgiving Holiday
	Fri, 11/29/2013		Thanksgiving Holiday
36	Mon, 12/02/2013		Student presentations I
37	Wed, 12/04/2013		Student presentations II
38	Fri, 12/06/2013		Student presentations III
	Mon, 12/09/2013		Begin Take-home final

No Title PHY 711 - Assignment #1

08/28/2013

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

b= κ
tan(θ/2)
.

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On 28 Aug 2013, 00:48.

No Title PHY 711 - Assignment #2

8/30/2013

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.

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On 29 Aug 2013, 23:21.

PHY 711 -- Assignment #3

Sept. 2, 2013

Finish reading Chapter 1 in Fetter & Walecka.

In the last lecture, we derived the differential cross section for the elastic scattering of two hard spheres -- dσ/dΩ|_CM( θ) = D²/4, where D is the sum of the radii of the two spheres. Now suppose that in lab frame of reference, the incident mass (m₁) has an initial velocity v₁ and the target mass (m₂) is at rest.
- Find the relationship between the center of mass scattering angle θ to the lab frame scattering angle χ.
- Find the relationship between the lab and CM differential cross sections.
- Evaluate these expressions for the case that m₁=m₂. Check your results to make sure that the total scattering cross section is the same in the two frames of reference.
Extra credit: Work Problem 1.16 in Fetter and Walecka

PHY 711 -- Assignment #4

Sept. 4, 2013

Read Chapter 2 in Fetter & Walecka.

Complete the analysis of the Foucault pendulum, using either the method used in the lecture notes or the method used in the text. Estimate the pendulum period for Wake Forest University which is located at lattitude 36 deg and longitude -80 deg.

PHY 711 -- Assignment #5

Sept. 6, 2013

Start reading Chapter 3 in Fetter & Walecka.

Work problem 3.10 at the end of Chapter 3 of Fetter & Walecka.

PHY 711 -- Assignment #6

Sept. 11, 2013

Continue reading Chapter 3 in Fetter & Walecka.

Work problem 3.5 at the end of Chapter 3 of Fetter & Walecka.

No Title PHY 711 - Assignment #7 PDF version

Note: This problem is similar to one posed in the Classical Mechanics text by Goldstein.

9/13/2013

Continue reading Chapter 3 in Fetter and Walecka.

Consider a Lagrangian function which depends on ·· q(t) in addition to q(t), · q(t) and t; L=L(q,· q,·· q;t).
1. Show that the Euler-Lagrange equation for this Lagrangian is:
  
  d²
  dt²
  ⎛
  ⎜
  ⎝ ∂L
  ∂
  ⋅⋅
  
  q
  
  ⎞
  ⎟
  ⎠ − d
  dt
  ⎛
  ⎜
  ⎝ ∂L
  ∂
  ⋅
  
  q
  
  ⎞
  ⎟
  ⎠ + ∂L
  ∂q
  = 0.
2. Consider the specific Lagrangian given below and find the corresponding equations of motion.
  
  L(q,
  ⋅
  
  q
  
  ,
  ⋅⋅
  
  q
  
  ;t) = − m
  2
  q
  ⋅⋅
  
  q
  
  − k
  2
  q².

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On 10 Sep 2013, 21:51.

No Title PHY 711 - Assignment #8

9/16/2013

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 15 Sep 2013, 22:51.

PHY 711 -- Assignment #9

Sept. 18, 2013

Continue reading Chapters 3 and 6 in Fetter & Walecka.

Work problem 6.5 at the end of Chapter 6 of Fetter & Walecka.

PHY 711 -- Assignment #10

Sept. 20, 2013

Continue reading Chapters 3 and 6 in Fetter & Walecka.

Reconsider problem 6.5 at the end of Chapter 6 of Fetter & Walecka. Write the Hamiltonian and Hamilton's equations of motion for this system. Also determine the constants of motion.

PHY 711 -- Assignment #11

Sept. 23, 2013

Continue reading Chapters 3 and 6 in Fetter & Walecka.

This problem demonstrates the Liouville Theorem for a simple case of one dimensional motion of particles of mass m subject to a constant force F. Suppose that at time t=0, the range of the initial positions are 0 ≤ x₀ ≤ Δx and the range of initial momentum are 0 ≤ p₀ ≤ Δp so the initial phase space area is Δx Δp .
- Determine the Hamiltonian for this system and the canonical equations of motion.
- Write explicit trajectories for x(t), p(t) for the 4 extreme initial conditions: (x₀, p₀) = (0,0), (Δx,0), (0,Δp), (Δx,Δp).
- Estimate the phase space area for some time t > 0 using the trajectories found above.

PHY 711 -- Assignment #12

Sept. 25, 2013

Continue reading Chapter 6 in Fetter & Walecka.

Setup and fully solve the problem posed in problem 6.5c in first the Lagrangian formalism and then the Hamiltonian formalism. You may assume the initial conditions are
1. r(t=0)=0.
2. v(t=0)=V_x0 x + V_y0 y.

PHY 711 -- Assignment #13

Sept. 27, 2013

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. 30, 2013

Finish reading Chapter 6 in Fetter & Walecka.

Read parts of the paper by Hans C. Andersen "Molecular dynamics simulations at constant pressure and/or temperature" in which he constructs a Lagrangian function to represent a system of particles held at constant pressure α.

Starting with the Lagrangian function (3.2), derive the Hamiltonian function (3.6).
Derive the equations of motion (3.7).

PHY 711 -- Assignment #15

Oct. 4, 2013

Continue 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 #16

Oct. 7, 2013

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.

PHY 711 -- Assignment #17

Oct. 14, 2013

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

Oct. 16, 2013

PHY 711 - Homework # 18

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 14 Oct 2013, 03:26.

PHY 711 -- Assignment #19

Oct. 18, 2013

Finish reading Chapter 7 in Fetter & Walecka

Find the Fourier series coefficients for the periodic function considered in Lecture 22 (slide 10)
In the interval -T ≤ t ≤ T: f(t) = t/T
Note that for this function, the period is 2T.

No Title

Oct 21, 2013

PHY 711 - Problem Set # 20

Continue 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 2012, 12:32.

No Title

Oct 23, 2013

PHY 711 - Problem Set # 21

Continue 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 24 Oct 2012, 12:50.

PHY 711 -- Assignment #22

Nov. 06, 2013

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.

PHY 711 -- Assignment #23

Nov. 08, 2013

Continue reading Chapter 9 in Fetter & Walecka.

Consider the analysis for the speed of sound in an ideal gas as we derived in class. 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 #24

Nov. 11, 2013

Continue reading Chapter 9 in Fetter & Walecka.

Work problem 9.3 at the end of Chapter 9 in Fetter & Walecka.

PHY 711 -- Assignment #25

Nov. 13, 2013

Continue reading Chapter 9 in Fetter & Walecka.

Read Appendix C and D in Fetter & Walecka. Starting with Eq. C.15 and change of variables C.16, "derive" Eq. C.18. Convince yourself that Eq. D2.17 is a solution to Eq. C18. Also show that Eq. C.23 follows from Eq. C.13.

PHY 711 -- Assignment #26

Nov. 15, 2013

Finish reading Chapter 9 in Fetter & Walecka.

In class, we analyzed the one-dimensional motion of an adiabatic ideal gas for the case of the traveling wave with wave velocity u=|v|+|c|. Derive the functional form of u for the case u=|v|-|c|. Using the maple script (or an equivalent with your favorite software), visualize the density wave form at t=0 and t=1.

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Last modfied: Wednesday, 20-Nov-2013 01:03:30 EST