1. 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(3) and h(3).

PHY 711 -- Assignment #2

Read Chapter 1 in Fetter & Walecka.

• In class, we started the derivation of the differential cross section for the elastic scattering of a beam of particles of mass m having an initial velocity u_i hitting a stationary spherical hard sphere target having mass M (uniformly distributed) with mutual radius R, scattered at an angle θ in the laboratory frame of reference. Complete the derivation to find the expression for the differential cross section as a function of R.

PHY 711 -- Assignment #3

Read Chapter 1 in Fetter & Walecka.

• Work out the details of the equations relating the laboratory scattering angle to the scattering angle in the center of mass reference frame. Similarly, work out the details of the relationship between the differential scattering cross sections in the laboratory and center of mass frames.

PHY 711 -- Assignment #4

Continue reading Chapter 1 in Fetter & Walecka.

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

PHY 711 -- Assignment #5

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 #6

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=5).

1. 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.

1. 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.

1. 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 #10

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

Start reading Chapter 4 in Fetter & Walecka.

1. 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

Finish reading Chapter 4 and start Chapter 7 in Fetter & Walecka.

1. 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'.

1. At t=0,
   

   μ(x,0) = A cosh(x)       and       ∂μ(x,0) ∂t =0,

   (2)

   where A is a given amplitude.
2. At t=0,
   

   μ(x,0) = 0      and       ∂μ(x,0) ∂t = A sinh(x) cosh2(x) .

   (3)

PHY 711 -- Assignment #14

Continue reading Chapter 7 in Fetter & Walecka.

Consider the Sturm-Liouville equation with τ=1 and σ=1 for the interval 0 ≤ x ≤ 1 and the boundary values df(0)/dx=df(1)/dx=0.

• Find the lowest eigenvalue and the corresponding eigenfunction.
• Choose a reasonable trial function to estimate the lowest eigenvalue and compare the estimate to the exact answer.

1. Assume that a > 0 and b > 0; use contour integration methods to evaluate the integral:
   
   

   ⌠ ⌡ ∞ 0  cos(ax) x4+4b4 dx.

PHY 711 -- Assignment #16

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 ≡ 3m and m₃=m₄ ≡ m.

1. Evaluate the moment of inertia tensor for this object in the given coordinate system.
2. Find the principal moments of inertia and the corresponding principal axes. Sketch the location of the axes.

PHY 711 -- Assignment #17

Continue reading Chapter 5 in Fetter & Walecka.

1. Work problem 5.10 at the end of Chap. 5 of Fetter & Walecka.

PHY 711 -- Assignment #18

Read Chapter 8 in Fetter & Walecka.

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

PHY 711 -- Assignment #19

Read Chapter 9 in Fetter & Walecka.

1. Consider the example discussed in class on slides 7-10 concerning the flow of an incompressible fluid in the z direction in the presence of a stationary cylindrical log oriented in the y direction. Consider the case where the log is replaced by a stationary sphere. Find the velocity potential for this case, using the center of the sphere as the origin of the coordinate system and spherical polar coordinates.

PHY 711 -- Assignment #20

Continue reading Chapter 9 in Fetter & Walecka.

1. 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 #21

Finish reading Chapter 9 in Fetter & Walecka.

1. Review the derivation of Eq. 52.53 and discuss its limiting values.

PHY 711 -- Assignment #22

Start reading Chapter 10 in Fetter & Walecka.

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

PHY 711 -- Assignment #23

Start reading Chapter 11 in Fetter & Walecka.

1. 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.