In one sense, the starting point for this article is Feynman’s Lost Lecture on Newton’s Principia. In another sense, the starting point is the Principia itself, and the mystery surrounding Newton’s attempt to explain Kepler’s First Law without the use of the calculus that he himself invented. Did Newton actually succeed in this attempt? Even a genius such as Feynman could not follow Newton’s arguments in this regard. So, instead, he created a brilliant geometrical argument of his own, culminating in an elegant construction of an ellipse. My article will not further address this dramatic climax to Feynman’s Lost Lecture, but instead will build on his preceding arguments to develop a novel, quantitative, and formula-based proof of Kepler’s First and Third Laws, without recourse to the formal methodology of calculus. For viewers interested in Feynman’s geometrical demonstration, Grant Sanderson has created an exquisite graphical representation of Feynman’s coup de grace in a video you can find here on Youtube. For a more detailed overview of this series of video presentations, I recommend that you take a look at this synopsis.

In the part 2 video presentation, essential features of the geometry of ellipses will be reviewed, and the terminology required to study planetary orbits will be defined.

Newton’s proof of Kepler’s second law, as elucidated by Feynman, relies on Newton’s first 2 laws of motion, a few basic principles of geometry, and an intuitive concept of infinitesimals and limits. As it happens, the rate at which a planet sweeps out area during its orbit is equal to the specific angular momentum, h, divided by 2, where h equals the angular momentum of the planet divided by its mass. In mathematical terms, dA/dt = h/2.

If we assume that all orbits are ellipses, then we can imagine a circular orbit, because circles are a special case of ellipses. Although we might not be able to find a perfectly circular orbit anywhere in the universe, it can be quite fruitful to consider a hypothetical circular orbit as a thought experiment. In Part 4, we will prove that Kepler’s 3rd law is true for a circular orbit. Also, we will show that for a circular orbit, the speed of the planet equals the square root of GM/D, where D is the radius of the circle. This latter finding will be very useful later on.

When Feynman could no longer follow Newton’s reasoning in the Principia, he charted his own course, which began with establishing that the change in the speed of the planet equals the change in the angle of azimuth times a constant. As it turns out, that constant equals GM/h.

So, the magnitude of the dV vector for any fixed incremental change in azimuth is a constant, and, of course, its direction is always towards the sun. To visualize the significance of this statement, we consider the planet’s varying velocity at macroscopic intervals. We then return to the realm of infinitesimals to discover the true power of Feynman’s dV/d-theta circle

In order to gain greater familiarity with the dV/d-theta circle and how it can be used as a tool, we apply it here to the rather trivial case of a circular orbit.

Imagine that a planet in a circular orbit is “rear-ended” by an asteroid traveling in exactly the same direction but at a higher speed, causing an inelastic collision. In accordance with the law of the conservation of linear momentum, the magnitude of the planet’s velocity would increase, but its direction would not. How will its orbit change?

Building on everything that we have learned so far, we set out to find the equation for the orbit of a planet, given its position and velocity at perihelion.

We can employ several of the formulas we derived previously to prove Kepler’s Third Law with just a handful of algebraic manipulations.

You are given the speed and distance from the sun of planet Q, and also the angle between its velocity and displacement vectors, at a particular moment as it orbits the sun, and nothing else. Can you find the equation describing its orbital path? You will need a knowledge of trigonometry, the geometry of ellipses, and much of the material covered earlier in this article. Extra credit (solution not provided): How long does it take planet Q to travel from perihelion to the point where its speed and distance from the sun first matches those given as “initial conditions”.