[Farsight (OP)]

I was reading PhysicsWorld this morning, and read an excellent article Dark Energy: how the paradigm has shifted. I followed a reference and also read Dark Energy: back to Newton?. This suggests that Newton had developed a lambda term. See http://www.homepages.ucl.ac.uk/~ucapola/CLrev.pdf and note this expression and quote:

F/m = r¨ = – GM /r² + CMr

"What is intriguing is that, having completed this discussion, Newton explores the consequences of a wide range of central force laws and comes to the conclusion that there is a second form for which spherically symmetric masses can be treated as if all the mass is located at the central point. That is when “the compounded force with which two spheres attract each other is as the distance between the centres of the spheres” (Newton 1687, Proposition 77, Theorem 37). He comments in the Scholium: “I have now explained the two principal cases of attractions: when the centripetal forces decrease as the square of the ratio of the distances, or increase in a simple ratio of the distances, causing the bodies in both cases to revolve in conic sections, and composing spherical bodies whose centripetal forces observe the same law of increase or decrease in the recess from the centre as the forces of the particles themselves do; which is very remarkable.”

Does anybody know anything about this?

[PhysBang]

Despite what Lahev and Calder say, it is not simply possible but highly probable that Newton was investigating Hooke's spring law formulation of celestial force in the passage outlined.

The big difference in the laws is that when we look at the force holding the Moon in orbit, if we think of it in terms of Newtonian gravity, it gives a measurement of the force of gravity that is almost exactly equal to that of measurements done on the surface of the Earth. If we think of the force holding the Moon in orbit as Hooke's force, then we get a measurement of a force that thas nothing to do, physically or numerically, with the force of Gravity on Earth. It is the measurement unity that is demonstrable from investigation that Newton uses in almost all of his arguments from investigation in Book III of the Principia.

Given Newton's reasoning, he could be quite happy in using the Moon test (or other tests) as an approximate measure of the force of what we call Newtonian gravity, and using deviations from what it might predict to measure the influence of the Hooke term. This is essentially what Einstein did in using the perihelion advance of Mercury as evidence for General Relativity.

I think that it is anachronistic to consider Hooke's force to be something like dark energy in concept, even though it shares some mathematical characteristics.

Other than this, the history in that article seems pretty good.

[Farsight (OP)]

Thanks PhysBang. I'll have to read up on this.