Inverse square law

[caledonia]

In the proofs of the inverse square law for planetary orbits, as deduced by Newton in the Principia from Keplers Laws, he assumes lots of properties of the conics. Mathematicians in his day were well versed in Geometry, including the Conics. Not so nowadays.

 

I started with Apollonius but found it tough going - many proofs are very long. Archimedes showed that an ellipse is also a section of a right cylinder, and this simplifies some theorems. Then again, Dandelin spheres are great for proving other properties. I found two other proofs myself, much easier than Apollonius, in the case of the parabola.

 

If anyone else would like to be able to follow Newton's geometrical proofs, I could post my documents (as a set of pdf's).

[Markus Hanke]

Why make it complicated ? Just model the gravitational field as going radially outward from an isolated point source, and with spherical symmetry ( a physically reasonable model ); this makes the field both irrotational and source-free outside the point source, in other words, it makes it conservative. Mathematically, for a vactor field F, this means that outside the source the following conditions are fulfilled :

 

[math]\displaystyle{\triangledown \cdot \mathbf{F}=0}[/math]

 

and

 

[math]\displaystyle{\triangledown \times \mathbf{F}=0}[/math]

 

So both the divergence and the curl must vanish. In flat 3-dimensional space this automatically implies an inverse square law, as can easily be shown in spherical coordinates from the above differential equations.

Edited May 3, 2013 by Markus Hanke

[Enthalpy]

Maybe Caledonia wants to understand it in a historical perspective.

[elfmotat]

Maybe Caledonia wants to understand it in a historical perspective.

 

The fast, easy version is that Kepler formulated his third law using the work/data of various people, relating the period of a planet's orbit to its orbital radius. Newton's law of gravitation was then a natural generalization which explained Kepler's law.