What is the mathematical proof for Kepler's first law?

[babbi.mamgain]

I wanted to ask how come kepler say that the orbits on which planet rotates is elliptical ... just by obvious saying, what we have learned is that when the velocity is Perpendicular to force the path traced is circular...even if we say in reality the actual angle B/W velocity and force is not 90 ...but i question is what makes a planet to go that far from the focci ? try to give an answer easy to understand. or please try to explain the mathematical proof in detail.

 

[HallsofIvy]

That's a very strange question!

just by obvious saying, what we have learned is that when the velocity is Perpendicular to force the path traced is circular... even if we say in reality the actual angle B/W velocity and force is not 90

I'm not sure what you mean by "by obvious saying". You appear to be saying that even if something is NOT true, we can say it is! That is certainly not true.

It is not a matter of something "forcing" a planet to move in a non-circular orbit. It is a matter of there not being anything to force the planet to go in a circular orbit! A circular orbit has "eccentricity" exactly equal to 0. An elliptic orbit can have eccentricity anywhere from 0 to 1. It would be very surprising if a parameter were exactly 0 rather than some range of values.