Math 642-600 Assignments — Spring 2021
Assignment 1 Due Wednesday, 2/3/2021
- Read section 5.1.
- Do the following problems.
- Problem 2, 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. The point is that one need not assume that
y′′ exists.)
- Smoothing spline. Let J[y] =
∫01 (y′(x))2dx. Let the
admissible set for J be all piecewise C1 curves on [0,1]
that satisfy y(k/n) = yk , for k = 0, 1, 2, ..., n, with
the discontinuities in y′ appearing only at the points
xk = k/n. Use
the previous problem to show that the minimizer for J is a linear
spline passing through all of the points {(xk,yk), k = 0,
..., n}. (Hint: y(x) is in C1[xk,
xk+1]).)
Assignment 2 Due Wednesday,
2/24/2021 Monday, 3/1/2021
- Read sections 5.2 and 5.4
- Do the following problems.
- Problem 6, page 205 (§ 5.2).
- Problem 7, page 206 (§ 5.2). (See Fig. 5.4, p. 205 for a
diagram.)
- Problem 8, page 206 (§ 5.2). (Note: There is an error in the
expression for L. The (dθ/dt)2
sin2θ) should be replaced by
(dφ/dt)2 sin2θ.)
- 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.)
- For a planet having mass $m$ in orbit around the sun (mass=$M$),
the potential is $V(r)=-mMG/r$, where $G$ is the gravitational
constant. In class, we showed that the Hamiltonian for this system is
$H=\frac{1}{2m} (p_r^2+r^{-2}p_\theta^2) - mMGr^{-1}$, and that $E=H$
and $\ell=p_\theta$ are constants of motion for the system. Inserting
these constants in $H$, we have
$E=\frac{1}{2m}\big(p_r^2+r^{-2}\ell^2\big)-mMGr^{-1}$.
- Show that $p_r = \ell r^{-2} \frac{dr}{d\theta} = - \ell
\frac{d}{d\theta}(r^{-1})$, if we change the independent variable from
$t$ to $\theta$.
- Let $u = r^{-1}- m^2MG\ell^{-2}$. Show that $2m E\ell^{-2} +
m^4M^2G^2\ell^{-4} = \big(\frac{du}{d\theta}\big)^2 + u^2$. To
simplify notation, let $\rho=(m^2MG\ell^{-2})^{-1}$ and
$\gamma=2m\rho^2 E \ell^{-2}$. This puts the equation above in the
form $u=r^{-1} -\rho^{-1}$ and
$\rho^{-2}(1+\gamma)=\big(\frac{du}{d\theta} \big)^2 +
u^2$. (Obviously, $\gamma+1>0$. For the orbit to be closed, one needs
$\gamma <0$. Consequently, $0<\gamma+1<1$.) Solve this differential
equation for $u$ in terms of $\theta$ and obtain $r=r(\theta)$.
- Use your solution to the previous part to establish Kepler's laws
of planetary motion:
- The orbit of a planet is an ellipse, with the sun at one
focus.
- The radius vector from the sun to the planet sweeps out equal
areas in equal times.
- The square of the period of any planet is proportional to the
cube of the semimajor axis of its orbit. The proportionality constant
is the same for all planets.
Assignment 3 - Due Monday, 4/25/2021.
- Read section 7.2.1 and 7.2.3.
- Do the following problems.
- Let $L$ be a bounded linear operator on $\mathcal H$; that is, $L
\in {\mathcal B}(\mathcal H)$. Show that $\rho(L)$ contains the
exterior of the disk $\{\lambda \in {\mathbb C} \colon |\lambda|
\le ||L||\}$. (Consequently, $\sigma(L)$ is contained in the disk.)
- Section 7.1: 2(a), 2(b). For each operator in 2(a), 2(b), find
the norm and adjoint $L^\ast$. (You may use problem 1 above.)
- Let $L=L^\ast$ and suppose that for every $f\in D_L$ we have that
$0\le \langle Lf,f\rangle \le 1$. Show that $\sigma(L)\subseteq
[0,1]$.
- Let $Lu=-u''$, $D_L=\{u,u''\in L^2[0,1]\}: u(0)=u(1)=0\}$.
- You are given that the spectrum of $L$ is
$\sigma(L)=\{n^2\pi^2\}_{n=1}^\infty$. Show that, for $\lambda
\not\in \sigma(L)\cup \mathbb R_{\le 0}$, the Green's function
for $L-\lambda I$ is given by
\[ G(x,y;\lambda)=-\frac{1}{\sqrt{\lambda}\sin(\sqrt{\lambda})} \begin{cases}
\sin(\sqrt{\lambda}(1-y))\sin(\sqrt{\lambda} x) & 0 \le x \le y \le
1\\ \sin(\sqrt{\lambda}(1-x))\sin(\sqrt{\lambda} y) & 0\le y \le x
\le 1\end{cases}\]
- Let $\sigma_n=\{n^2\pi^2\}$. Show that $P_{\sigma_n}f= b_n\sin(n\pi
x)$, where $b_n = 2\int_0^1 \sin(\pi y) f(y)dy$. (Hint: Use the
residue theorem to find $\int_C G(x,y\lambda)d\lambda$, where $C$ is a
simple closed curve with the point $\sigma_n$ inside.)
- Show that in $L^2[0,1]$ we have $f(x)=\lim_{N\to
\infty}\sum_{n=1}^N b_n \sin(n\pi x)$.
- Find the Green's function for the operator $Lu = -u''$, with
$D_L=\{u,u''\in L^2[0,\infty): u'(0)=0\}$. Use it and Stone's formula for the
spectral family of $L$ to derive the Fourier cosine transform pair:
\[
F(\xi)= \frac{2}{\pi}\int_0^\infty f(x)\cos(\xi x)dx \ \text{and}\
f(x)= \int_{0}^\infty F(\xi)\cos(\xi x)d\xi
\]
Updated 4/15/2021 (fjn)