Orbit of planet have to be a conic section

AI Thread Summary
The discussion centers on proving that planetary orbits are conic sections using geometry rather than differential equations. One participant expresses skepticism about the possibility of such a proof, suggesting that observations are the primary means of validation. Another contributor references a derivation from Halliday and Resnick related to circular orbits influenced by gravitational attraction, highlighting the relationship between the bodies' distances from their center of mass. They mention Newton's geometric proof in "Principia" but struggle to locate a downloadable version of the text. The conversation emphasizes the historical and theoretical aspects of understanding planetary motion.
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How we can prove (using geomtry and not differential equation) that the orbit of planet have to be a conic section.
 
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I don't think you can "prove" such a thing...except maybe by observations.

Please take that as an opinion, not a firm statement, as I have been away from such things a looooong time...I surprised I even remembed there was an example of a the derivation from a textbook...


My old version of Halliday and Resnick has a derivation closely linked to your query, but not so general...for CIRCULAR orbits of two bodies m and M under the influence of each others gravitational attraction.

It starts with a center of mass of the combined bodies such that mr = MR, where r,R are the radial distances of bdoies m,M, from the center of mass, and noting the centripetal forces of the two bodies equal and figuring the gravitational attraction equal results in

GMm/(R+r)2 = mw2r...
 


i saw in wikipedia that Newton prove it usinge geometry in his book "principia"
but i can't find download of the book/
 

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