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.