MATH 396 PROBLEMS 13

IGOR KRIZ

Regular problems:

1. Using the Lagrangian approach to mechanics, solve the motion of one particle in one-dimensional space with constant homogeneous force field. In other words, find a function y : [a, b] → ℝ with given y(a) and y(b) which minimizes the integral of “energy”

\frac{1}{2}m(y′)^{2} + mgy

with time ranging from a to b. [Tip: the “simplified” Euler equation in this case leads to a more complicated computation, but a more explicit answer.]

2. Prove that if f(x + iy) = f_{1}(x + iy) + if_{2}(x + iy) is a holomorphic function, f_{1}, f_{2} : ℂ → ℝ, then both f_{1} and f_{2} are harmonic, which means that

\frac{∂^{2}f_{1}}{∂x^{2}} + \frac{∂^{2}f_{1}}{∂y^{2}} = 0,

and similarly for f_{2}. [Use the Cauchy-Riemann conditions.]

3. Let U be a connected open set in ℂ. A function f has the property that

∫_{L} \frac{f(ζ)}{(ζ − z)^{2}} = 0

for every circle L with center z where L is contained in U. What can you say about f?

4. Find the geodesics on the cylinder {(x, y, z)|x^{2} + y^{2} = 1}. [Parametrize your curve as (x, y, z) = (cos θ, sin θ, u). Treat θ as an independent variable, u as a function of θ. Write the “simplified” Euler equation for the variation problem minimizing the integral of the derivative of length: (\frac{dx}{dθ}, \frac{dy}{dθ} , \frac{dz}{dθ}).] Is there an elementary explanation of your answer?

Challenge problem:

5. Determine the geodesics on the parabolloid {(x, y, z)|x^{2} + y^{2} = z}. [Parametrize (x, y, z) = (r cos t, r sin t, r^{2}).]