Orbit of planet have to be a conic section

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SUMMARY

The discussion centers on proving that planetary orbits are conic sections using geometric methods rather than differential equations. A reference is made to Halliday and Resnick's textbook, which provides a derivation for circular orbits of two bodies influenced by gravitational attraction. The derivation involves the center of mass and equating centripetal forces, leading to the equation GMm/(R+r)² = mw²r. Additionally, it is noted that Isaac Newton demonstrated this concept geometrically in his work "Principia."

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  • Understanding of gravitational forces and centripetal motion
  • Familiarity with the concept of center of mass
  • Basic knowledge of conic sections in geometry
  • Access to Halliday and Resnick's physics textbook
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  • Study the derivation of circular orbits in Halliday and Resnick's textbook
  • Research Isaac Newton's "Principia" for geometric proofs of planetary motion
  • Explore the mathematical properties of conic sections
  • Learn about gravitational interactions between multiple bodies
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Students of physics, educators teaching gravitational dynamics, and anyone interested in the geometric foundations of celestial mechanics.

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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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