I have recently developed an interest in understanding the life, times, and work of Isaac Newton. Reading a handful of books aimed at a general audience I learned a great deal, but realized that they gave little actual mathematical detail insight to his contributions. I have a PhD in mathematics, and have taught calculus, so I am certainly familiar with the modern version of the techniques he developed. But that is not the same thing as appreciating what it meant at the time. I have never had to connect anything in a calculus textbook to the subject of astronomy.
Newton, in his Principia Mathematica outlined a mathematical framework that gave excellent predictive power, and confirmed Keplar's thesis of elliptical orbits. I could have sought out a proof that Keplar's fundamental laws of planetary motion can be derived from Newton's calculus, but I felt that it might be more rewarding to see how it could be exhibited computationally. The following animation applies a simple discrete time approximation of Newton's laws to draw out an elliptical orbit of the "earth" about the "sun". You can see that the velocity changes, the earth moving faster as it draws closer to the sun. Note that nothing here is "to scale". I have simply implemented the inverse square law to produce an ellipse.
x = , y =
dx = , dy =