Math 121A: Homework 11-
1. (a) In polar coordinates, the total length of a curve r(θ) is given by the functional
I[r] = θ1∫θ2 ds = θ1∫θ2 √(r2 + (dr/dθ)2) dθ.
By using the Euler equation, derive a differential equation for geodesics in polar coordinates. (Note: the question asks you derive the differential equation, but not to solve it.)
(b) Consider the a straight line from the points (x, y) = (1, -1) to (x, y) = (1, 1). Write the line as a function r(θ) and show that it satisfies the differential equation from part (a).
2. (a) By using the Euler equation, derive a differential equation for the function y(x) the minimizes the distance
1∫2 g(x)ds = 1∫2g(x) √(1 + y'2) dx
where g(x) is an arbitrary function.
(b) Using the boundary conditions y(1) = 0 and y(2) = 2, solve the differential equation for the two cases of g(x) = 1 and g(x) = √x.
(c) Plot the two solutions y(x), and discuss how their shapes correspond to what is being minimized.
3. (a) For a function y(x) defined on an interval x1 ≤ x ≤ x2, consider the case of minimizing the functional
I[y] = x1∫x2 F(x, y, y', y'')dx.
By following the original Euler equation derivation, and assuming zero values of the variation η(x) and its derivative at x1 and x2, show that in this case the Euler equation becomes
(d2/dx2)(∂F/∂y'') - (d/dx)(∂F/∂y') + (∂F/∂y) = 0.
(b) Find the solution y(x) on the interval 0 ≤ x ≤ 1 that minimizes
I[y] = ½ 0∫1(y'')2 dx
subject to the boundary conditions y(0) = 1, y(1) = -1, and y'(0) = y'(1) = 0. Calculate the value of I[y].
(c) Consider the function y∗(x) = cos πx. Show that it satisfies the boundary conditions of part (b). Calculate the value of I[y∗] and compare its numerical value with I[y] from part (b).
4. Consider a particle of mass m sliding on a frictionless vertical hoop of radius a, whose position is described by the angle θ(t) that it makes with the negative y axis, so that (x, y) = (a sin θ, -a cos θ). Assume that gravity points in the negative y direction.
(a) Write down a Lagrangian for the system, in terms of the kinetic energy minus the potential energy, and hence derive a differential equation for θ(t) describing the particle's position.
(b) Solve the equation for the case when θ is small, corresponding to the case of small oscillations about the vertical axis. Find the frequency of oscillations.
(c) Optional for the enthusiasts. What physical principle does the Beltrami identity correspond to?