Mathematical framework for elliptical orbits

[daveed]

how do elliptical orbits work? in school, we never learned how to calculate the orbits of objects, their velocities, accelerations, etc. , we always worked with perfect circular orbits.

what is the mathematical framework behind elliptical orbits, then?

 

[Tide]

Elliptical orbits are intimately linked to the inverse square nature of the gravitational force. Even before Newton gave the complete picture, Kepler recognized the orbits of the planets are elliptical and his "equal areas in equal times" laws of planetary motion can be seen to relate to the principle of angular momentum conservation.

Finally, there's a good reason you usually work with circular orbits in school - it's a lot easier! I.e., the detailed mathematical analysis is a bit beyond beginning calculus.