Lesson 9
Calculus of Variations; Euler's Equation

1) Your everyday experience, common sense, and math classes all suggest to you that the shortest distance between two points in two-dimensional space is a straight line (unless, of course, you want to consider a warped space-time continuum!).

Please describe how you would go about proving that this is true, using the methods of variational calculus. Don't actually do this, but describe in English the procedure you would follow. Please be specific in your steps.

2) Have you heard about the new amusement park ride, the "Brachistochrone Slide"? Its slide ramp is in the shape of a cycloid, and is made out of space-age frictionless material. The slide begins at a high point and ends a few meters lower, at the lowest point of the cycloid. If you want the Brachistochrone Slide to take advantage of as much of the cycloid as possible (while still having the bottom of the slide at the lowest point of the cycloid), about how long should the distance x in the figure be?

Also, about how long (in time) will the Brachistochrone Slide ride take?

[Hints: See problem 6.6, which is one of your homework problems for next week. See also figure 6.4 in the text.]

3) Consider a piece of a flexible rope hanging freely between two points of support (the red and blue points in the figure). We can use variational calculus methods to determine what shape the rope hangs in.

What quantity do you think is minimized by the shape the rope actually hangs in in real life?

1. the tension in the rope
2. the time it would take for a particle to slide (without friction) along the shape from one endpoint to another
3. the average slope of the shape
4. the potential energy of the rope

You may change your mind as often as you wish. When you are satisfied with your responses click the SUBMIT button.

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