Reading: Chapter 10: pp. 192-195; Chapter 11: pp. 201-203.
Problem L: The (kinetic) energy of a space curve α(t) is defined as 1/2 of an integral, as follows:
E(α) = ½ ∫ab ||α'(t)||2dt.
Most of the following steps are elementary and require just one or two supporting statements.
(1) Show that the energy of a curve may depend on reparametrization of the curve (unlike the arc length, see Proposition 5.4).
(2) Show the inequality:
L(α)2 ≤ 2 (b-a)E(α),
where
L(α) = ∫ab ||α'(t)|| dt
is the arc length, and that the equality applies if and only if α is parametrized by a (constant) multiple of arc length. [Hint: Google out Cauchy-Schwarz for integrals.]
(3) Conclude that a curve between two points on a regular surface minimizes energy if and only if it has a minimal arc length and is parametrized by a multiple of arc length.
(4) Explain scientifically why cyclists prefer not to stop at a stop sign, even though whether they stop or not does not change the distance covered.
(5) Conclude that an energy-minimizing curve on a surface satisfies the geodesic equations. [Hint: You may use properties I have described on the News web page of April 5.]
Problem M: Show that a parallel x(u0,t) on a surface of revolution is a geodesic if and only if the radius λ (u) has a critical point at u0. [Note that the converse of Clairaut's Relation does not apply here.]
Problem N: Find a formula for the exponential map on the unit sphere.
Last modified: (2013-04-09 14:06:53 CDT)