Nature of Laws of Physics (Gravity)

[V0ODO0CH1LD]

I am going to explain what I am looking for, of course all the while hoping that what I am looking for actually exists, by talking about how I got comfortable with the idea of harmonic oscillators.

I never quite understood why the forces on a harmonic oscillator were of the form $F=-kx$, I just accepted that it was an empirical fact. People measured infinitely many springs and all kinds of oscillating "things" in laboratories and by some awesome coincidence they all turned out to obey this one law. Nature was compliant.

It wasn't until I read this one article that I realized that things were not so simple. The whole idea of $F=-kx$ coming from taylor series approximations of the potential energy was fascinating! It is not a coincidence that all things that oscillate around their state of equilibrium behave the same. It's just that they can all be reduced to the same approximation.

So that is what I am looking for with gravity. And what I've been trying to look for in everything else since the harmonic oscillator insight. It's hard to believe that two things like gravity and electricity have force laws so similar. Did someone really run enough experiments that the data just screamed $F=G\frac{m_1m_2}{r^2}$? Or is there another perspective?

And in general; are most laws of physics achieved by the same "technique" as the harmonic oscillator force law? As in, it makes sense theoretically, let's see if nature rolls with it?

 

[DrStupid]

Newtons law of gravitation was based on the experiments of Galileo Galilei and the astronomical observations of Tycho Brahe (summarized by Johannes Kepler in his laws of planetary motion).

 

[V0ODO0CH1LD]

Okay, but how did they arrive at the equation? It's hard to believe that they tried different expressions until one agreed with the experiments..

 

[mfb]

Yes, scientists ran experiments and the data screamed F=GmM/r².

And in general; are most laws of physics achieved by the same "technique" as the harmonic oscillator force law? As in, it makes sense theoretically, let's see if nature rolls with it?

I would expect that the observation preceeded the theoretical explanation. Sometimes it is like that, sometimes it is the other way round.

It's hard to believe that they tried different expressions until one agreed with the experiments..

It is easy if the function is easy, scientists do that frequently.

 

[DrStupid]

It's hard to believe that they tried different expressions until one agreed with the experiments..

To my knowledge Kepler did it this way. Of course his choice was limited by the astronomical data but there still was a huge number of possible expressions to test. He was close to go insane that time.

Newton just needed to apply his laws of motion to the orbits to get the force acting on the planets.

 

[HallsofIvy]

Well, it was famously Sir Isaac Newton who showed that Kepler's laws required an "inverse square law".

 

[vlado_skopsko]

Gravitational and electric forces obey the inverse square law because of the way the energy spreads. Its purely geometric thing. The energy equipotentials are spheres and they have inverse square dependence with the radius.

 

[SteamKing]

This article gives a little discussion of Kepler's laws and their derivation mathematically:

http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion

Copernicus had postulated his heliocentric theory in an attempt to simplify Ptolemy's theory of planetary motion (which had complicated devices known as epicycles). As more planetary bodies were discovered, Ptolemy's theory could not explain the motions of these bodies.

Tycho Brahe had compiled a chest full of observational data on the movement of the planets, but he had no theories which explained why the planets moved as they did. When Brahe died, Kepler, who had been working for Brahe, took the chestful of data with him when he took up his next job.

Kepler wanted to explain planetary motion and was aware of Copernicus' heliocentric theory. It took Kepler 10 years to come up with the first two of his laws of planetary motion. It took a further 10 years of work for him to discover the third law.

At approximately the same time as Kepler was devising his laws, Galileo and others were trying to explain planetary motion using the gravitational attraction of the sun on the various planets. No one could determine with the mathematics of the day if gravity could explain why the planets moved as they did.

Edmund Halley (who was an astronomer by profession) went to see Newton at Cambridge (this was many years after the deaths of both Galileo and Kepler) and asked Newton what sort of orbit would be produced in a planet which was acting under the gravity of the sun where the gravitational force obeyed the inverse square law. Halley was rather surprised when Newton immediately answered, "An ellipse, of course!" Halley further inquired how Newton knew this, and Newton said he had calculated it but he had misplaced his calculations. Halley further pressed Newton for his calculations, which Newton promised to recreate and send to Halley. It was out of this meeting with Halley that the Principia was eventually written by Newton to explain how gravity acting according to an inverse square law led directly to Kepler's laws.

 

[technician]

I never quite understood why the forces on a harmonic oscillator were of the form F=−kx, I just accepted that it was an empirical fact. People measured infinitely many springs and all kinds of oscillating "things" in laboratories and by some awesome coincidence they all turned out to obey this one law. Nature was compliant.

Nature is simple, not compliant

First of all... you never have to 'accept that it is an empirical fact'...empirical facts are facts ! they don't care whether you accept them or not !
The forces on harmonic oscillators are not all of the form F = -kx, This is the form of the SIMPLEST harmonic oscillator, known as SIMPLE HARMONIC MOTION.
The fact is that the simplest (mathematically) to analyse is when F is proportional to displacement. It would be equally desirable to analyse the motion that results when F is proportional to x² or when F is proportional to x^1/3 etc.

 

[WannabeNewton]

Instead of the simplicity of nature being hard believe, I would like to think that the grand beauty of nature is exactly in its innate simplicity and elegance (until you actually start doing computations!).