Math 642-600 Assignments
Assignment 1 - Due Wednesday, 1 February.
- Read section 5.2, 5.4
- Do the following problems.
- Find the Legendre transformation H(p) for F(x) =
ex. Use your favorite software (mine is MATLAB) to graph y
= F(x); show the envelope of tangents. Do the same for H(p).
- 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.
- Let L(q1,...,qn,
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.)
- 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.
- Problem 6, page 208 (§ 5.2).
- Problem 7, page 208 (§ 5.2).
Assignment 2 - Due Wednesday, 8 February.
- Read section 6.1, 6.2
- Do the following problems.
- Problem 4, page 208 (§ 5.2).
- Problem 1, page 209 (§ 5.4).
- Problem 3, page 210 (§ 5.4).
- Problem 4, page 210 (§ 5.4).
- Calculate the second variation of a functional of the form
J[y] = ∫ab F(x,y′)dx.
Show that this functional has no conjugate points.
- 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.
- Problem 7, page 277 (§ 6.1) (Joukowski's map).
- Problem 6, page 278 (§ 6.2).
- Problem 12, page 278 (§ 6.2).
- Problem 13, page 278 (§ 6.2) (Assume α ≥ 2).
- Problem 1, page 278 (§ 6.3).
- Problem 5, page 279 (§ 6.3).
- 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.
- Problem 3, page 281 (§ 6.5).
- Problem 5, page 282 (§ 6.5).
- Problem 10, page 282 (§ 6.5).
- Problem 11(b), page 282 (§ 6.5).
- Problem 15, page 283 (§ 6.5).
- Problem 22, page 283 (§ 6.5).
- Show that the area ωn of the unit sphere
Sn in Rn+1 is given by
ωn = 2π(n+1)/2/Γ((n+1)/2).
Hint: find the integral of exp(− x12
− ... − xn+12 ) over
Rn+1 in two ways, using Cartesian coordinates and
using ``spherical coordinates.''
Assignment 5 - Due Wednesday, 19 April.
- Read sections 8.3, 10.1
- Do the following problems.
- Problems 8, 9, 10, pg. 328 (§ 7.2)
- Problems 4(d), 7, 10, pg. 329 (§ 7.3)
- Let Hn(t) be the degree n Hermite polynomial. Show
that Hn(t)e− t2/2 is an
eigenfunction of the Fourier transform. Find the corresponding
eigenvalue.
- Let f be in Schwartz space, and let g be C∞ and satisfy
|g(j)(t)| ≤
cj(1+t2)nj
for all nonnegative integers j. Here cj and nj
depend on g and j. Show that fg is in Schwartz space.
- Consider the piecewise quadratic
T(t) = (t+1)2(H(t + 1) − H(t + ½)) + (½
− t2)(H(t + ½) − H(t − ½))
+ (t-1)2(H(t − ½) − H(t − 1)),
which is 0 outside of [−1,1]. Find the Fourier transform of T
this way. Find the distributional derivative T(3), which is
a linear combination of δ functions. Then, find the Fourier
transform of this tempered distribution. Finally, use Fourier
transform properties to obtain the desired transform.
Assignment 6 - Due Wednesday, May 3.
- Do the following problems.
- Problems 2, 4, pg. 391 (§ 8.2)
- Problem 5, pg. 455 (§ 10.2)
- Problem 11(a), pg. 456 (§ 10.3)
- Problems 8, 9, pg. 457 (§ 10.4)
Updated 4/9/06 (fjn).