Math 642-600 Assignments — Spring 2008
Assignment 1 - Due Wednesday, January 23.
- Read sections 4.1-4.3, 5.1.
- Problem 2, page 172 (§ 4.1).
- Problem 4, page 172 (§ 4.1).
- Problem 5, page 172 (§ 4.1).
- Problem 10, page 172 (§ 4.1).
- Problem 3, page 173 (§ 4.2).
- Problem 7, page 173 (§ 4.2).
- Problem 12, page 173 (§ 4.2).
- Problem 2, page 174 (§ 4.3).
- Problem 3, page 174 (§ 4.3).
Assignment 2 - Due Wednesday,
February 6.
- Read sections 4.1-4.3, 5.1.
- Do the following problems.
- Problem 2, page 204 (§ 5.1).
- Problem 6, page 204 (§ 5.1).
- Problem 7, page 204 (§ 5.1).
- 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.
- Consider the functional J(y) =
∫abF(x,y,y′)dx,
where y ∈
C1 and y(a)=A, y(b)=B are fixed.
- 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 = ∫ab
(-g(x) +
Fy′(x,y,y′))η′(x)dx,
where g(x) =
∫axFy(u,y,y′)du.
- Use (a) to show that -g+Fy′(x,y,y′)=c,
a
constant. From this, it follows that
Fy′(x,y,y′)∈ C1.
(Du
Bois-Reymond, 1879.)
Assignment 3 - Due Wednesday, February 20.
- Read sections 5.2, 5.4.
- Do the following problems.
- 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) = 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.
- Suppose that the kinetic energy of a system is given by
T=∑i,j
mi,jqi′qj′, where
mi,j= mi,j(q1,
... qn). The matrix with entries mi,j is
symmetric and positive definite. Also, let the potential energy of the
system be U(q1, ... qn). 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, pk =
∂L/∂qk′, k = 1, ..., n. (Hint: use the
answer to the previous question to do this problem.)
- 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).
- 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.
- Problem 1, page 207-8 (§ 5.4).
- Problem 4, page 208 (§ 5.4).
- 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?
- Problem 8, page 208 (§ 5.4).
- Problem 4, page 274 (§ 6.1).
- Problem 5(a,c), page 275 (§ 6.2).
- Problem 6, page 276 (§ 6.2).
Assignment 5 - Due Friday, April 4.
- Read sections 7.1 and 7.2.
- Do the following problems.
- Problem 6, page 279 (§ 6.5).
- Problem 12, page 280 (§
6.5). (See hint.)
- Problem 20, page 281 (§ 6.5)
- 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 Wednesday, April 16.
- Read sections 7.3.3, 7.5-7.5.1.
- Do the following problems.
- Show that if L is self adjoint, then its spectrum is a subset of
the reals.
- Problem 2(a,b), page 328 (§ 7.1)
- Problem 1(b), page 329 (§ 7.2) (You may use the Green's
function that you found in problem 1 of the midterm.)
- Problem 5, page 329 (§ 7.2)
- Problem 6(a,c), page 329 (§ 7.2)
- Problem 7, page 329 (§ 7.2)
- Problem 11, page 330 (§ 7.2)
Assignment 7 - Due Friday, April 25.
- Read sections 10.1 and 10.3.
- Do the following problems.
- Let Hn(x) be the degree n Hermite polynomial. Show
that Hn(x)e− x2/2 is an
eigenfunction of the Fourier transform. Find the corresponding
eigenvalue. (Hint: use the formula Hn(x) =
ex2(d/dx)n e−
x2 for the Hermite polynomials.)
- 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.
- Problem 3(d), page 333 (§ 7.5)
Updated 4/19/08 (fjn).