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Introduction You have probably learned, perhaps repeatedly, the canonical equation of Newtonian physics: F = ma.
You have probably learned, perhaps repeatedly, the canonical equation of Newtonian physics: F = ma. This equation underlies essentially every dynamical system, from the interaction between atoms to the movement of entire galaxies. Sometimes, when we’re talking about a particularly simple solution, we can use this simple rule to write out the trajectory of an object. For example, if we throw a ball through the air and ignore air resistance, we know that its height as a function of time can be represented as:
Our solar system is, of course, not just the Sun and the Earth, so let’s also add in the Moon as well. The Moon is quite small compared to these other two bodies, so we expect that it shouldn’t have much impact on the trajectories. And this is true, but it turns out the effect it has is, while small, monumental in what it does to our ability to solve for the solution. That’s because as soon as we add the Moon—our third body—there no longer exists a closed-form solution for the trajectories of any of the three objects.
Since we can’t write out an analytic expression for the trajectory anymore, we instead turn to numerical methods, simulating the trajectory one time step at a time to construct the resulting trajectory. We’re going to learn more in the second half of the course about how to utilize some built-in tools in MATLAB to numerically simulate such a system, but for the midterm project, you’re going to create a rudimentary solver and use it to simulate a model of the motion of the Sun, Earth, and Moon. It may sound complicated, but we will tell you step by step what you need to do.
You can find on Gauchospace a file called “3body.mat.” Loading the file in MATLAB will provide you with five 1 × 3 vectors, corresponding to values for three different objects (so the first value in each vector is for the first object, etc).
Using these provided values, you will be responsible for simulating the system. You will do this by writing two functions and a script, the details of which can be found below. The files you will be expected to submit are:
Finally, Section 4 of this homework asks you to answer some questions. Please submit this as a PDF file generated by Word in addition to your Matlab files. If you don’t have Microsoft Word on your computer, not to worry. It is easy to create a Word document on Box. Instructions on how to do this are included for download in this midterm project on GauchoSpace.