Deriving Newton's Law of Gravitation

[lavinia]

Given Kepler's Laws how do you derive Newton's Law of Gravitation?

 

[AUK 1138]

I don't believe any of Kepler's laws were used to create Newton's law. Newton added an expression to one of Kepler's laws to allow it to work in standard SI units as opposed to only years/ AU, but i think that's where the "teamwork" ends. I could be wrong, though.

 

[lavinia]

I don't believe any of Kepler's laws were used to create Newton's law. Newton added an expression to one of Kepler's laws to allow it to work in standard SI units as opposed to only years/ AU, but i think that's where the "teamwork" ends. I could be wrong, though.

but what if I wanted to do the derivation?

According to Wikipedia Huygens had formulated F=Ma before Newton and had also described the law of centripetal force for circular motion (http://en.wikipedia.org/wiki/Christiaan_Huygens#Mechanics).

Maybe we could throw those in along with Kepler's Laws.

 

[Cleonis]

Given Kepler's Laws how do you derive Newton's Law of Gravitation?

The inverse square law of gravity addresses the physics on a deeper level than Kepler's third law does law does.

I think F=ma was already common knowledge before Galileo. It was the law of required centripetal force for circular motion that wasn't common knowledge. As you mention, Huygens was the first to demonstrate the magnitude of the required centripetal force. (His demonstration was not elegant or efficient, later Newton devised a clear derivation.)

As a first approximation the planetary orbits can be taken as circular. Quite possibly it was Hooke who first worked out the following: if all the planets of a solar system have perfectly circular orbits, then with an inverse square law of gravity the periods of the orbits will follow Kepler's third law.

While that is compelling evidence for an inverse square law of gravity, it doesn't constitute a satisfactory derivation. The actual orbits are ellipses, as described by Kepler's first law.

It was Newton who provided a derivation for the case of the actual ellipse-shaped orbits.
(Actually, only a few of Newton's contemporaries had the capabilities to follow all the steps of Newtons geometrical derivations. Newton's results became accessible to the scientific community when they were rederived using calculus. In Newton's time his results were well known and understood, but not his methods.)


Interestingly, it is not necessary to use the law of required centripetal force.
Kepler's first law states that every planet orbits in an ellipse, with the Sun at one focus of that ellipse.
There is in fact only a single force law that gives rise to such orbits: the inverse square law.

The best way to proceed is probably as follows:
- Demonstrate that the inverse square law of gravity gives rise to Kepler orbits.
- Demonstrate that this relation is unique. That is, demonstrate that it's a one-on-one relation; that other force laws will give rise to other orbits.

Then the uniqueness proves it the other way round: that Kepler orbits imply an inverse square law.

 

[JDługosz]

Hmm, how about this:

Kepler's equal-area law is a statement of conservation of angular momentum, which implies that the force is between the planet and the sun.

Supposing the orbit to be circular, Kepler's distance/time relation implies that the acceleration is inverse square. Since the planet's are presumably different masses, that implies that force is proportional to the mass of the planet.

Running with that hypothesis, start with an inverse square law and see what that gives you. You find that solutions include circles, ellipses, parabolas, and hyperbolas. Realizing that this is wonderful in that it let's you immediately drop the earlier simplifying assumption (rather than treating this as a first approximation), conjecture that the planet's orbits are ellipses with the sun at a focus, and go get some observational data to confirm.

As I recall from an episode of NOVA, some other scientist asked Newton what he thought the shape would be if there were an inverse square law. He replied that he didn't have to guess but knew it would be an ellipse, and feigned not knowing where his notes were to show him, since he didn't want to let anyone know that he had a secret weapon (the calculus).

 

[lavinia]

going from inverse square to ellipse is classical.

the other way around seems hard. I agree that equal area in equal time seems like a good starting place.

 

[Cleonis]

going from inverse square to ellipse is classical.

the other way around seems hard. I agree that equal area in equal time seems like a good starting place.

Well, Kepler's second law, that equal areas are swept out in equal intervals of time, is equivalent to conservation of angular momentum.

Conservation of angular momentum applies for any central force, not specifically for the inverse square law of gravity. So you can't bring the area law to bear on this question; the area law doesn't single out the inverse square law.

 

[lavinia]

Well, Kepler's second law, that equal areas are swept out in equal intervals of time, is equivalent to conservation of angular momentum.

Conservation of angular momentum applies for any central force, not specifically for the inverse square law of gravity. So you can't bring the area law to bear on this question; the area law doesn't single out the inverse square law.

so what do you do?

 

[Cleonis]

Given Kepler's Laws how do you derive Newton's Law of Gravitation?

Kepler's first law suffices.

A Google search with the keywords "gravity" "kepler" "integrate" "ellipse" gave several promising links, among which a .pdf called
http://www.math.uconn.edu/~troby/Math200S08/kepler.pdf" .

As you can see, working it out takes considerable mathematical skill.

Try reading several authors; among mathematicians it's quite a popular exercise, by the looks of it. What the demonstrations have in common is that at some point in the proceedings an expression is integrated. That integration is the crucial step.

What you start with is an expression for the amount of acceleration that will occur (the inverse square law) and the expression for that is integrated over time to obtain the resulting trajectory.

In the demonstrations polar coordinates are used. In the last step it is pointed out that the obtained expression for the resulting trajectory is the formula for a conic section.

 

[Cleonis]

Given Kepler's Laws how do you derive Newton's Law of Gravitation?

A Google search with the keywords "gravity" "kepler" "integrate" "ellipse" gave several promising links,

The following article is very interesting:
http://arxiv.org/ftp/arxiv/papers/0812/0812.2755.pdf"

The authors use geometric means only, similar to the kind of presentation that Newton used in the Principia.
Interestingly, Kepler's second law, the area law, plays a part in their derivation. The contribution of the area law is that allows time to be expressed geometrically.