Proof of the inverse proportionality of R^2 to the attraction force

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Discussion Overview

The discussion centers around the proof of the inverse proportionality of the square of the distance between two objects to the gravitational attraction force between them. Participants explore theoretical and mathematical perspectives, including geometric interpretations and historical context related to Newton's Law of Gravitation.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning
  • Historical

Main Points Raised

  • Some participants propose a heuristic approach involving the force spreading out on the surface of an expanding sphere, suggesting that this leads to an inverse square law under Euclidean geometry.
  • One participant expresses a belief that the relationship between inverse-square laws and three spatial dimensions is fundamental, linking it to symmetries and conservation laws.
  • Another participant questions the validity of the inverse square law, suggesting that if it does not hold, the explanation must lie in the physics rather than mathematics.
  • A participant challenges the notion that the inverse-square law has been mathematically proven, indicating that it is primarily supported by empirical measurements.
  • There is a mention of Newton's demonstration that the inverse square law is the only force law that results in elliptical orbits for planets, which some participants reference as a historical point of interest.

Areas of Agreement / Disagreement

Participants express differing views on the mathematical proof of the inverse-square law and its implications. There is no consensus on whether the law has been proven mathematically or if it is solely based on empirical evidence. Additionally, the interpretation of geometric principles and their relevance to the law remains contested.

Contextual Notes

Some participants note that the discussion may depend on definitions of geometry (Euclidean vs. non-Euclidean) and the assumptions underlying the inverse-square law. The conversation also touches on the historical context of gravitational theory without resolving the mathematical aspects.

parsa418
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Could anyone tell me how to exactly prove that the distance between two objects squared is inversely proportional to the attraction force between them?
Thanks.
 
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A heuristic approach is to think of the force spreading out on the surface of an expanding sphere. In order for the total force to stay constant, it needs an inverse square law, as long as the geometry is Euclidean.
 
mathman said:
A heuristic approach is to think of the force spreading out on the surface of an expanding sphere. In order for the total force to stay constant, it needs an inverse square law, as long as the geometry is Euclidean.

I'm glad you posted this- that's exactly how I was taught to think of the relationship. That is, the relationship between inverse-square laws and 3 spatial dimensions is as fundamental as the relationship between symmetries and conservation laws.

However, when I said this to a colleague in the Math Department a few weeks ago (in the context of experiments looking for extra dimensions, for example

http://adsabs.harvard.edu/abs/2007gras.conf...9P
http://www.springerlink.com/content/l64187120j67q780/

), he scoffed, said "No, that's not right", and wandered off. Was he referring to non-Euclidean geometry, could he have thought of something else, or was he just giving me a hard time?
 
I can't read his mind! If it does not obey an inverse square law, then the explanation has to be in the physics, not mathematics.
 
Andy Resnick said:
Was he referring to non-Euclidean geometry, could he have thought of something else, or was he just giving me a hard time?

Well I think it's fairly elementary that a field radiating symmetrically outwards in N euclidian dimensions decays like 1/r^(N-1). Pick any part of what I just said that you could possibly poke holes in and perhaps that's what he meant... Euclidian geometry is usually the one people bring up most often, though...
 
Has the inverse-square law of gravity ever been proven in a mathematical sense? I thought it was only "proven" in the sense that measurements confirm it.
 
The way Newton showed that gravity obeys an inverse square law was toshow that that is the only force law that results in elliptic orbits for the planets.
 

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