So I have been pondering about Newton lately. Mostly because I heard various stories that might be apocryphal. I could not find a reference to them, but they strike me as being true. There is a legend about apples falling on Isaac Newton’s head as a story of how he discovered the law of gravitation…

Of course, this is probably just a fancy legend concocted after the facts to paint a more romantic picture of the discovery. What is true however, is that Newton had some hint of using a central force to explain the motion of the planets from Hooke. However Hooke could not solve the problem, and Newton had to invent calculus and differential equations to really solve this problem.

These ideas of gravitation have had a profound impact on modern science. The idea that the laws of motion that govern the planets are the same as the laws of motion that pertain to us has become a standard definition of what physics is about: the laws of physics are universal and should apply to everything and everyone all over the known universe. We have a lot of evidence that this is so. That whatever can happen here, given the same circumstances, will happen elsewhere in pretty much the same way.

The other thing that was introduced was an ‘unpleasant’ action at a distance, whereby one seems to communicate force without rods or any obvious physical agency to transmit the information of where objects sit. This problem was solved later by Einstein, but that is a discussion for another day. Of course, the law of gravitation was able to predict the experimentally observed laws of Kepler to very good accuracy.

With the modern tools, it does not take too long to prove Kepler’s laws for the motion of the planets, given the universal law of gravity. That law is that the force of attraction of bodies is proportional to the product of the masses and inversely proportional to the square of the distance between them.

The minus sign indicates it is an attractive force. Since the force is central, one can prove that angular momentum is conserved. In this case, one has an equation of the form

where $\mu$ is the so called reduced mass, and are a polar coordinate system. This is the equation that shows that one sweeps equal areas in equal times. For infinitesimal times one has a triangle of base , and height .

One can also use the conservation of energy

and the conservation of angular momentum to convert the equation of energy conservation into

where the new variable has been substituted. This equation looks like a harmonic oscillator equation of motion for the variable (shifted from the origin) in terms of the ‘time’ described by the angle $latex\theta$, and the general solution is of the form

if one defines carefully. With a little bit of manipulation one can show that this is the equation describing an ellipse. I’ll leave it to you as homework.

Now, it turns out that I heard a story recently about why it took Newton so long to publish his results. Astronomers in his time had observed the distance from the earth to the moon and gave it a size. Newton’s law of gravitation predicted that size, and the astronomical observation was off by a factor of three…

When it was later calculated again ( a few years later) it agreed with Newton’s derivation and after that Newton published his results.

I have not been able to cross check this story out there, and I would appreciate it if someone can give me some indication that this is true or false. To me, it has a ring of truth to it.

on November 10, 2008 at 6:45 amzeynelHowever Hooke could not solve the problem, and Newton had to invent calculus and differential equations to really solve this problem.I am not sure if professional historians would agree, but as far as I know Newton did not have “to invent calculus and differential equations” to solve planetary orbits.

In the Principia Newton does not use equations or calculus to do astronomical calculations. Newton works strictly with proportions. Consequently, Newton did not know Newton’s constant of gravitation either.

Here are the relevant propositions from the Principia where Newton computes astronomical quantities. They are all simple algebraic manipulations of Kepler’s rule.

on November 10, 2008 at 11:41 amCVOne way to try to find out whether that has any substance is reading Subrahmanyan Chandrasekhar’s book about Newton’s principia.

I never read it all but the parts I read had many sourced remarks about the things you just mentioned.

on November 10, 2008 at 4:00 pmUncle AlNoether’s theorem couples conservation of angular momentum to vacuum isotropy. Lab and cosmic observations prohibit EM vacuum anisotropy as aether, Lorentz violation, and chiroptical effects (time-reversal even, imaginary part of the complex gyrotropy tensor). Calorimetry constrains

massed sectorvacuum anisotropy below 10^(-12) relative.Covariance with respect to reflection in space and time is not required by the Poincaré group of Special Relativity or the Einstein group of General Relativity. A parity Eötvös experiment detects 5×10^(-14) relative massed sector vacuum anisotropy. The Equivalence Principle may suffer parity violation. Chemically identical, opposite parity atomic mass distributions then do not conserve angular momentum.

Single crystals in enantiomorphic space groups P3(1)21 and P3(2)21 are extreme opposite parity atomic mass distributions. Cultured alpha-quartz provides parity Eötvös test masses. Benzil, mp 95 C, provides parity calorimetry test masses.

Somebody should look.

on November 10, 2008 at 6:08 pmSvenI attended a talk on the derivation of this law once, actually. According to the professor giving the talk, Newton derived the law using calculus, but I don’t believe he ever published that. Later, when he was writing the Principia, he had adopted a serious favouritism for the geometrical proofs, and so proved it geometrically and algebraically. Apparently there was some confusion over whether or not his proof was accurate, too; I gather it’s a bit arcane.

on November 11, 2008 at 1:50 amdberensteinHi Sven:

I have heard the same. This is one of the aspects that makes reading the Principia really hard. I have leafed through it and did not find it illuminating. I would suspect that Newton was trying to give proofs in a language that his peers were already familiar with.

on November 11, 2008 at 6:43 amzeynelOne way to try to find out whether that has any substance is reading Subrahmanyan Chandrasekhar’s book about Newton’s principia.

Yes, that’s an overwhelming book. I think he spent like ten years writing it.

The problem with that approach to Principia is that Book 1 and Book 3 are independent of each other. To understand how Newton computes astronomical quantities in the Principia you don’t need to know anything more than how to do simple algebraic manipulations.

If the Principia were 20 pages long the astronomical content would not have suffered. When I tried to read Chandrasekhar’s book I did not yet know this so I started but couldn’t finish it. Then I found Dena Densmore’s Newton’s Principia: the central argument. That’s a great book because she just concentrates on the central argument, as the title says.

And it is thel opposite of Chandrasekhar’s approach. Chandrasekhar is totally anachronistic, Densmore declares in the Introduction as her principle that she will use no anachronistic notion whatsoever. This is great because then you actually understand what Newton was doing because she fills in all the details that Newton left out to make the book hard to read. As mentioned above Newton was writing for his peers. I believe that this is the way to read the Principia. I would recommend Densmore’s book to anybody who is interested in the Principia.

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