Math 642-600 Assignments — Spring 2010
Assignment 1 - Due Wednesday, 2/3/2010.
- Read sections 5.1.
- Do the following problems.
- Section 4.3: 3, 4, 6
- Section 4.5; 12.
- Section 5.1: 2, 5, 6
- 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 so that its energy is a minimum,
find the chain's shape.
- Let x = (x,y) and let J[x] =
x2y(x2 + y2)− 1
if x ≠ 0, J[0] = 0. Show that the
Gateâux derivative exists at at x = 0 for every
direction η = (h,k). Also, show that the Frechét derivative
doesn't exist at x = 0.
Assignment 2 - Due Friday, 2/12/2010.
- Read sections 5.2.1, 5.2.2, and 5.4.
- Do the following problems.
- Show that if y = F(x) is a convex, C2
function on [0,1], then H(p), the Legendre transform of F, satisfies
H(p) = maxx ∈ [0,1] (xp - F(x))
- Find the Legendre transformation H(p)
for F(x) = xT A x,
where x is in Rn and A is a
symmetric, positive definite n×n matrix.
- Consider a system for which the Lagrangian L(q1,...,qn,
q′1,...,q′n) = T−V has the form
L = ½(q′)TG(q)q′ − V(q1,...,qn).
where G is a symmetric, n×n matrix that depends only on the qj's, and q′ is a column vector containing the time-derivatives of the qj's.
- Show that as as function of the p's, H = ½pTG−1p + V(q1,...,qn). Here, p is a column vector containing the pj's. Briefly explain why the first set of Hamilton's equations below hold:
q′j = ∂H/∂pj, j = 1, ..., n.
- Show that ∂(G−1)/∂qj = − G−1(∂G/∂qj)G−1.
- Use the results above to show that the rest of Hamilton's equations hold:
p′j = − ∂H/∂qj, j = 1, ..., n.
- In class we showed that the Hamiltonian H =
(pr2 + r −
2pθ2)/(2m) − K/r for a planet
in orbit around the sun. In addition, we also found that H = E and
pθ = L are constants of the motion.
- Show that pr = Lr − 2dr/dθ =
−L d(r − 1)/dθ.
- 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.
- Use your solution to the previous part to show that the orbit is
an ellipse.
- Problem 7, page 206 (§ 5.2).
Assignment 3 - Due Friday, 2/19/2010.
- Read sections 6.1 and 6.2
- Do the following problems.
- Section 5.4: 1, 3, 4, 8
- Section 5.4: 6. Take Ω to be a 2D disk centered at 0
and having radius r = a. How does the lowest
eigenvalue change with the radius a?
- Use a quadratic polynomial to estimate the lowest eigenvalue for
the Sturm-Liouville problem with Lu = − u′′ =
λu, u(0) = 0, u′(1)+u(1) = 0.
- Section 6.1: 4
Assignment 4 - Due Friday, 3/5/2010.
- Read sections 6.4 and 6.5
- Do the following problems.
- Find the Laurent expansions for f(z) = ((z2 −
1)(z+2))−1 about z = 0.
- Section 6.2: 5(a,c,d), 6, 9
- Section 6.4: 2, 4, 17
Assignment 5 - Due Monday, 4/12/2010.
- Read sections 7.1 and 7.2
- Do the following problems.
- Section 6.5: 3(a), 6, 8
- Section 7.1: 1, 2(a,b)
- 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 −
tp), where tp 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.)
- Show that one may uniquely solve τ = u − ε
sin(u) for u = u(τ), and that u is an odd function of τ.
- 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 bn sin(nτ).
- Show that bn = (2/n)Jn(nε), n = 1,
2, ..., where Jn is the nth order Bessel
function of the first kind. Thus, we have that
u = τ + ∑n (2/n)Jn(nε)
sin(nτ).
Assignment 6 - Due Monday, 4/26/2010.
- Read sections 7.2.1, 7.2.2, 7.3.1
- Do the following problems.
- Show that if L is self adjoint, then its spectrum is a subset of
the reals.
- Show that if L is a bounded operator having norm ||L||, then
ρ(L) contains the disk |λ| > ||L|||.
- Section 7.2: 1(b) (You may use the Green's function in problem
2 from Test 1, take-home part.) 5, 6(b,c), 7.
Assignment 7 - Not to be turned in.
- Read sections 10.1-10.3
- Do the following problems.
- Let f be in Schwartz space, and let g be C∞ and
satisfy
|g(j)(x)| ≤
cj(1+x2)nj
for all nonnegative integers j. Here cj and nj
depend on g and j. Show that fg is in Schwartz space. Explain how to
define the product g(x)T(x), where T is a tempered distribution.
- Let T be a tempered distribution. Find the Fourier transforms for
T′, xT(x), T(x−a), and eibxT(x).
- Let T(x) := (1 - |x|)+, which is a "hat"
function. Explain why this is a tempered distribution. Find T′
and T′′, along with their Fourier transforms. Use these to
find the Fourier transform of T.
- Section 10.3: 1, 4, 7, 9, 11
Updated 5/2/2010 (fjn).