Newtonian Mechanics – Can Kepler’s Laws Derive Newton’s Gravitation Formula?

Question

I know that Newton's universal law of gravitation can be used, along with his laws of motion, to derive Kepler's laws. But what about the other way around? Can Kepler's laws be used to derive Newton's universal law of gravitation?

Answer 1

Yes, but it is somewhat contrived. You need to have defined angular momentum in the Newtonian way, too.

The fact that planets orbit in ellipses with the sun at one of the foci (Kepler's first law) can be made mathematical by just geometry, $$r(\theta) = \frac{p}{1+\varepsilon \cos\theta}$$ where $p$ is the semi-latus rectum, $\varepsilon$ is the eccentricity, $\theta$ is the angle from perihelion, and $r$ the radial distance from planet to star.

Just as you go from Newton's law to deriving the elliptic orbit in the usual way, you could now do this backwards to see what kinetic energy and potential could produce this path under assumption of Newton's 2nd law.

Instead of integrating a differential equation, differentiate this ellipse equation twice wrt $\theta$. This shows $$\ddot{r} + r = 1$$ Now somehow knowing to make a substitution like $u = \frac{1}{r}$, you could work backwards to an expression for the total energy and hence infer a law for the gravitational potential, and finally Newton's law of gravitation.

But note understanding which part of the total energy expression is due to gravity would require an understanding of angular momentum.