There are two common falsities of the history of gravity which are purported by school teachers:
- That Newton, upon being hit by a renegade apple, found the equation traced out by the seeds inside the fruit which, as he immediately realized, describes the motion of the planets.
- That Newton postulated the law of gravity and then used it to proved Kepler’s laws of planetary motion.
The first falsity is ridiculous and is rejected by anyone with half-a-brain. The second falsity, however, is held dear by many people even at the University level. This is not true.
Our freshman physics books may include a statement such as “Kepler’s laws of planetary motion can be derived from Newton’s laws of gravity.” Some of them even derives Kepler’s laws. Of course, they are not lying. However, they leave the impression that Isaac Newton did precisely that. That is, he started with his law of gravity that he found in his apple and then went to prove Kepler’s laws of planetary motion.
In truth, Newton went full circle. That is, he went from Kepler’s laws to his inverse-square law to Kepler’s laws again.
Don’t believe me? Read the damn Principia. I mean, it has the proof right there in Book I! Newton’s inverse square law was treated as a corollary with Kepler’s laws treated as a postulate.
The geometric proof Newton used is summarized well in Makers of Mathematics by Stuart and Hollingdale. The proof is entirely geometric which utilizes the geometric properties of ellipses to show that the attractive acceleration is indeed related to seperation by an inverse-square law.
A more palatable analytical proof can be found in Introduction to Celestial Mechanics by Moulten. It uses only calculus to prove the inverse-square law of acceleration. However, it is still pretty ugly so I don’t feel like putting it in this blog either (this blog will only have pretty proofs). However, the book is now in public domain and could be freely obtained online.
Both the geometric and analytical proofs share a similar property: they both do not utilize Kepler’s third law. That is, the square of the orbital period of a planet is proportional to the cube of the size of its orbit. This could be easily proven by Newon’s law of gravity. Hence, the third law is a corollary of the first two. However, in Mechanics by Landau, it was shown that, through a scaling argument, that Newton’s law of gravity can be derived from Kepler’s third law of planetary motion. However, this assumes that the Lagrangian is given by
(This is actually a corollary of Newton’s second law and Kepler’s second law). Landau shown that Kepler’s third law forces the potential energy describing gravity to be an inverse law of separation. This implies that the force, defined by the spatial derivatives of the Lagrangian, is inverse square.
This means that Kepler’s third and first laws of planetary motion are equivalent! [This assumes that we interpret Kepler’s third law as a scaling property of spacetime]. When one of them is paired up with Kepler’s second law, we obtain Newton’s inverse-square law. However, it is well known that Kepler’s second law of planetary motion, also known as the law of areas, is equivalent to the statement that the acceleration vector of the planet is parallel to the position vector of the planet with respect to the Sun. However, it is well known that the statement that all forces be central is actually equivalent to Newton’s first law, second law, and the conservation of energy (central forces are the only Galilean-invariant forces that obeys the conservation of energy).
Therefore, we can derive Newton’s law of gravity from a number of ways:
- Step I: Pick one of the following postulates: (A) Kepler’s first law . (B) Kepler’s third law.
- Step 2: Pick one of the following postulates: (A) Kepler’s second law. (B) Motion is described by central forces. (C) Newton’s first and second law with the conservation of energy.
- Step 3: Mix your choices for Step I and Step II, let it cool for an hour, and viola: You just made yourself an inverse-square law of gravity. Serves a family of four.
Of course, Kepler’s laws are a bit stronger than Newton’s inverse-square law since the latter admits hyperbolic orbits (parabolic and linear orbits are unstable). However, this is because it can be shown that Kepler’s laws also restrict a planet’s position in position-velocity phase space such that its energy relative to the Sun is negative so that its orbit is bounded. By making phase space unrestricted (except, of course, at the center), we obtain the possibility of hyperbolic orbits from Newton’s law of gravity.
What I want you to get out of this post is that Kepler’s laws of planetary motion and Newton’s law of gravity are very nearly equivalent. More generally, a lot of claims in physics are equivalent so be wary of people who say that A implies B but not the other way around.