Math 642-600 Assignments

Assignment 1 - Due Wednesday, 1 February.

• Read section 5.2, 5.4
• Do the following problems.
  1. Find the Legendre transformation H(p) for F(x) = e^x. Use your favorite software (mine is MATLAB) to graph y = F(x); show the envelope of tangents. Do the same for H(p).
  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. Let L(q₁,...,q_n, q′₁,...,q′_n) be a Lagrangian for a mechanical system. As we did in class, again derive Hamilton's equations for the system, but this time without any restrictions on the kinetic energy. (In class, we assumed that T depended only on the q′'s and not on the the q's.)
  4. A planet in the Sun's gravitational field has potential energy U(r) = − c/r, where r is the distance from the Sun to the planet and c is a constant. Find the planet's Lagrangian and Hamiltonian, in spherical coordinates. Bonus: Show that the orbit of the planet lies in a plane that includes the Sun.
  5. Problem 6, page 208 (§ 5.2).
  6. Problem 7, page 208 (§ 5.2).

Assignment 2 - Due Wednesday, 8 February.

• Read section 6.1, 6.2
• Do the following problems.
  1. Problem 4, page 208 (§ 5.2).
  2. Problem 1, page 209 (§ 5.4).
  3. Problem 3, page 210 (§ 5.4).
  4. Problem 4, page 210 (§ 5.4).
  5. Calculate the second variation of a functional of the form
     J[y] = ∫_a^b F(x,y′)dx.
     Show that this functional has no conjugate points.
  6. This problem is aimed at showing that there is no solution to the Jacobi equation that is strictly positive on an interval containing conjugate points. Consider the ordinary differential equation (ODE)
     -(P u′)′ + Qu= 0,
     where P and Q are C², and P(x)>0 for a ≤ x ≤ b. Suppose that solution the u₀ vanishes at x = a and x = b, but nowhere between. Show that if u is any other solution that is linearly independent of u₀, then there is a point x = c, a< c < b for which u(c) = 0. (Hint: consider the Wronskian of the two solutions u₀ and u.)

Assignment 3 - Due Wednesday, 22 February.

• Read section 6.3, 6.4, 6.5
• Do the following problems.
  1. Problem 7, page 277 (§ 6.1) (Joukowski's map).
  2. Problem 6, page 278 (§ 6.2).
  3. Problem 12, page 278 (§ 6.2).
  4. Problem 13, page 278 (§ 6.2) (Assume α ≥ 2).
  5. Problem 1, page 278 (§ 6.3).
  6. Problem 5, page 279 (§ 6.3).
  7. Find all possible Laurent expansions for f(z) = z(z+1)^-1(z-2)^-1 about the point z = 1.

Assignment 4 - Due Friday, 3 March.

• Read section 7.1, 7.2
• Do the following problems.
  1. Problem 3, page 281 (§ 6.5).
  2. Problem 5, page 282 (§ 6.5).
  3. Problem 10, page 282 (§ 6.5).
  4. Problem 11(b), page 282 (§ 6.5).
  5. Problem 15, page 283 (§ 6.5).
  6. Problem 22, page 283 (§ 6.5).
  7. Show that the area ω_n of the unit sphere Sⁿ in Rⁿ⁺¹ is given by
     ω_n = 2π^(n+1)/2/Γ((n+1)/2).
     Hint: find the integral of exp(− x₁² − ... − x_n+1² ) over Rⁿ⁺¹ in two ways, using Cartesian coordinates and using ``spherical coordinates.''

Assignment 5 - Due Wednesday, 19 April.