* Animation of Newton’s Mechanics *
To be a giant, you need a broad shoulder for others to climb on. Newton is such a figure. After ~400 years, we still have not exhausted his work†. Here is a version of his celestial mechanics that we can do without even bothering with differential equations. We know that Newton’s laws lead to celestial orbits. It’s easy to explain when every orbit is circular. The key point is that celestial orbits are not circular but elliptical. So we’re going to start from circles and see how they end up with ellipses as the answer.
1. Take 2 circles, r = 1 and r = cosΘ, one inside another. (It should look horizontal but excuse my tool. I use Excel so the graph appears vertically.) As time goes by, both circles span the same length, even though one is twice as large. These 2 circles will be what we base on to explore celestial mechanics.
2. Add these 2 circles point by point and we get a cardioid (red).
3. Take the reciprocal of the cardioid and we get a parabola (purple). Note that taking a reciprocal doesn’t lose the nature of the equation. Parabola is the ultimate celestial motion; it’s a one-time orbit where its eccentricity is 100%. The orbits of a comet are close to that.
4. If we add the 2 circles with a percentage (different eccentricity ε as in r = 1 + ε×cosΘ) before taking the reciprocal, we get various ellipses. Bingo! We’ve come up with a layman’s version of Newton’s mechanics.