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Computer Use Assignment

Kris Green: Math 223, Section 2

Fall 1998: Due Friday, August 28

The following function shows up in celestial mechanics:

$$z = f_\mu (x,y) = x^2 + y^2 + \frac{2(1-\mu)}{\sqrt{(x+\mu)^2+y^2}} + \frac{2\mu}{\sqrt{(x-1+\mu)^2+y^2}}$$

where $\mu$ is a parameter that is fixed. The physical situation (not necessary for working the problem) is that a large mass, say the earth, is located at $(-\mu,0)$ and a smaller mass, say the moon, at $(1-\mu,0)$.The coordinates are chosen so that the center of mass of the two objects is at (0,0) and the coordinates rotate around the center of mass at the proper rate. The function then gives the set of points where a third mass that is very small (say a satellite) would have zero orbital velocity. The function is used to find stable orbits for various situations.

The Problem: Start with $\mu = 0.8, -2 \le x \le 2, -2 \le y \le 2$and choose $0 \le z \le 8$ to view the function above in WinPlot. Note that this function is way too complicated to visualize without some sort of help.

1.

Look at the surface from a few different viewpoints. Try to describe, in words, what you see.

2.

Use WinPlot (under ``Mode'') to plot the level curves of the function. (At some point, you'll have to press ``Q'' to stop the plot.) Sketch these level curves. Be sure to include some values for the function along a few of the curves. You can print the level curves that the program generates, but make sure to add labels, by right clicking on the location and typing the text in.

3.

Does the computer accurately show the graph of $f_\mu$ near the points $(1-\mu,0)$ and $(-\mu,0)$? Why or why not? [Hint: look at the algeraic form of the function.]

4.

What effect does changing $\mu$ have on the surface? [For example, try $\mu = 0.4$ or 0.2.]

Getting Help: I will hold my office hours in Math 226 (the open access computer lab) for the rest of the week. This should allow you sufficient time to play with WinPlot and get an idea of what it can do for you.