Math 642-600 Assignments — Spring 2008

Assignment 1 - Due Wednesday, January 23.

• Read sections 4.1-4.3, 5.1.
  1. Problem 2, page 172 (§ 4.1).
  2. Problem 4, page 172 (§ 4.1).
  3. Problem 5, page 172 (§ 4.1).
  4. Problem 10, page 172 (§ 4.1).
  5. Problem 3, page 173 (§ 4.2).
  6. Problem 7, page 173 (§ 4.2).
  7. Problem 12, page 173 (§ 4.2).
  8. Problem 2, page 174 (§ 4.3).
  9. Problem 3, page 174 (§ 4.3).

Assignment 2 - Due Wednesday, February 6.

• Read sections 4.1-4.3, 5.1.
• Do the following problems.
  1. Problem 2, page 204 (§ 5.1).
  2. Problem 6, page 204 (§ 5.1).
  3. Problem 7, page 204 (§ 5.1).
  4. Hanging chain problem. A chain having uniform linear density ρ hangs between the points (0,0) and (1,0). The total mass m, which is fixed, and the total energy E of the chain are
     
     Assuming that the chain hangs in a shape that minimizes the energy, find the shape of the hanging chain.
  5. Consider the functional J(y) = ∫_a^bF(x,y,y′)dx, where y ∈ C¹ and y(a)=A, y(b)=B are fixed.
     1. Let y extremize J, subject to all of the conditions above. Verify that if η ∈ C¹ and η(a)=η(b)=0, then
        Δ J = ∫_a^b (-g(x) + F_y′(x,y,y′))η′(x)dx, where g(x) = ∫_a^xF_y(u,y,y′)du.
        
     2. Use (a) to show that -g+F_y′(x,y,y′)=c, a constant. From this, it follows that F_y′(x,y,y′)∈ C¹. (Du Bois-Reymond, 1879.)

Assignment 3 - Due Wednesday, February 20.

• Read sections 5.2, 5.4.
• Do the following problems.
  1. Show that if y = F(x) is a convex, C² function on [0,1], then the H(p), the Legendre transform of F, satisfies
     H(p) = max_{x ∈ [0,1]} (xp - F(x))
     
  2. Find the Legendre transformation H(p) for F(x) = x^T A x, where x is in Rⁿ and A is a symmetric, positive definite n×n matrix.
  3. Suppose that the kinetic energy of a system is given by T=∑_i,j m_i,jq_i′q_j′, where m_i,j= m_i,j(q₁, ... q_n). The matrix with entries m_i,j is symmetric and positive definite. Also, let the potential energy of the system be U(q₁, ... q_n). Find the Hamiltonian for the system and show that it has the form H = T + U, where the kinetic energy T is expressed in terms of the momenta, p_k = ∂L/∂q_k′, k = 1, ..., n. (Hint: use the answer to the previous question to do this problem.)
  4. In class we showed that the Hamiltonian H = (p_r² + r ^{− 2}p_θ²)/(2m) − K/r for a planet in orbit around the sun. In addition, we also found that H = E and p_θ = L are constants of the motion.
     1. Show that p_r = Lr ^{− 2}dr/dθ = −L d(r ^{− 1})/dθ.
     2. Let u = 1/r − mK/L. Show that 2m E/L² + (mK/L²)² = (du/dθ)² + u². Solve this first order differential equation for u in terms of θ.
     3. Use your solution to the previous part to show that the orbit is an ellipse.
  5. Problem 7, page 206 (§ 5.2).
  6. Problem 8, page 206 (§ 5.2). Also, find the Hamiltonian of the system, and write out Hamilton's equations for it. Finding integrals (constants) of the motion should follow immediately from these equations. (Note there is an error in the Lagrangian L. The second (dθ/dt)² on the right side should be (dφ/dt)².)

Assignment 4 - Due Friday, February 28.

• Read sections 6.1, 6.2, and 6.4.
• Do the following problems.
  1. Problem 1, page 207-8 (§ 5.4).
  2. Problem 4, page 208 (§ 5.4).
  3. Problem 6, page 208 (§ 5.4). Work with n = 2, and Ω a disk centered at 0 and having radius r = a. How does the lowest eigenvalue change with the radius a?
  4. Problem 8, page 208 (§ 5.4).
  5. Problem 4, page 274 (§ 6.1).
  6. Problem 5(a,c), page 275 (§ 6.2).
  7. Problem 6, page 276 (§ 6.2).

Assignment 5 - Due Friday, April 4.

• Read sections 7.1 and 7.2.
• Do the following problems.
  1. Problem 6, page 279 (§ 6.5).
  2. Problem 12, page 280 (§ 6.5). (See hint.)
  3. Problem 20, page 281 (§ 6.5)
  4. A planet moving around the Sun in an elliptical orbit, with eccentricity 0 < ε < 1 and period P has time and angle related in the following way. Let τ = (2π/P)(t − t_p), where t_p is the time when the planet at perihelion -- i.e., it is nearest the Sun. Let θ be the usual polar angle and let u be an angle related to θ via
     (1 − ε)^1/2 tan(u/2) = (1 + ε)^1/2 tan(θ/2).
     It turns out that τ = u − ε sin(u). All three variables θ, u, and τ are measured in radians. They are called the true, eccentric, and mean anomalies, respectively. (Anomaly is another word for angle.)
     1. Show that one may uniquely solve τ = u − ε sin(u) for u = u(τ), and that u is an odd function of τ.
     2. Show that g(τ) = u(τ) − τ is a 2π periodic function of τ. Show that the Fourier series of g(τ) is a sine series. That is,
        g(τ) = ∑_n b_n sin(nτ).
        
     3. Show that b_n = (2/n)J_n(nε), n = 1, 2, ..., where J_n is the n^th order Bessel function of the first kind. Thus, we have that
        u = τ + ∑_n (2/n)J_n(nε) sin(nτ).
        

Assignment 6 - Due Wednesday, April 16.

• Read sections 7.3.3, 7.5-7.5.1.
• Do the following problems.
  1. Show that if L is self adjoint, then its spectrum is a subset of the reals.
  2. Problem 2(a,b), page 328 (§ 7.1)
  3. Problem 1(b), page 329 (§ 7.2) (You may use the Green's function that you found in problem 1 of the midterm.)
  4. Problem 5, page 329 (§ 7.2)
  5. Problem 6(a,c), page 329 (§ 7.2)
  6. Problem 7, page 329 (§ 7.2)
  7. Problem 11, page 330 (§ 7.2)

Assignment 7 - Due Friday, April 25.