Hey guys! So I'm looking at this textbook that attempts to describe to me how to get kepler's first law describing a two-body system in polar coordinates, and I have no clue how to get from one step to another because I have seen NO OTHER PLACE solve this problem this way.(adsbygoogle = window.adsbygoogle || []).push({});

It starts off by describing angular momentum as L=m*r^2*w=m*r^2*do/dt and the total energy of the system as E=1/2mv^2-GMm/r. So far so good!

But then out of nowhere, it pulls out that our velocity is v^2=(dr/dt)^2 + (r*do/dt)^2. I'm guessing the first component is the radial velocity and the second is the veolcity relating to the circular motion of the body?? But why is everything squared? Shouldn't it just be v=dr/dt + rdo/dt????

Then I'm supposed to go through and replace do/dt using our trusty angular momentum equation and v using our energy equation to get rid of the time component and just have the diff eq dr/do = r sqrt((2mE/L^2)r^2 + (2GMm^2/L^2)r -1), which I worked through and got correctly.

And somehow, by sheer magic, I am supposed to solve this differential equation to get r=p/1+ecos(o + c). How so I solve this differential equation?? Eek!

So basically my questions are: How the heck does v^2=(dr/dt)^2+(r*do/dt)^2, and how the heck do I solve the resulting differential equation??

My brain would much welcome some help :)

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# Solving the Two-Body System

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