For the project # 2, unlike for the previous ones, you have a choice of topics. Please pick one from the following list and email me about your choice. Please do it no later than on April, 25. You will be expected to present a 15 minutes description of the project to the class at the very end of the semester. You are welcome to work in (small) groups.
In your presentation, you should include a number of relevant pictures and graphs, preferably (but not necessarily) produced by Maple. These pictures should be used to illustrate your exposition, not merely included without comment or reason (remember: ``a picture is worth a thousand words, but a thousand pictures are worthless''). Please pay attention to clarity of exposition; do not merely hand in an annotated Maple worksheet. While including relevant Maple commands is useful, your goal is to explain what you are doing from a mathematical point of view, not to describe how to use Maple to perform a certain task. You need not (although you can) state or prove the relevant theorems, but you should explain how you are using them to do your analysis.
In class, we have been discussing a system of differential equations which approximately models the flight of a glider. In this project, you are to explore a similar system, which models such a glider with a propeller or small engine attached. In this case, our model is
Here, as in class, v>0 is the speed of the glider through the air (not the horizontal speed), and is the angle the nose makes with the horizontal direction. Note that we have fixed the drag coefficient to be 0.3, but have added an additional term k to account for acceleration caused by the propeller.
You are to analyze and classify the solutions of this system for all (note that k=0 was covered in class). This means that you should find the various ranges of k where the behavior is qualitatively different.
Such analysis should include a discussion of the existence of fixed points (equilibrium solutions) and their linearizations (i.e., a discussion of eigenvalues and eigenvectors and how this relates to the solutions), as well as a discussion of the long-term behavior of the solutions.
In addition to the description of the trajectories in the -plane, you should also relate the solutions to properties of glider flight. For example, discuss whether the glider eventually must crash or can stay aloft indefinitely, whether the glider does loops, etc.
Go to http://www.math.com/students/wonders/life/life.html and learn the rules of the Game of Life. Implement the game with Maple. Show the behavior of a few interesting objects of your choice (but do include at least Gliders and Guns!).