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Kepler's second law, about equal areas in equal times, is a differential equation: it gives velocity as a function of location.

Where are the best expository accounts of the process of solving this equation, giving position as a function of time?

Michael Hardy
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    Are you looking more for a historical account of the problem and the experimental evidence which supported the law? Or are you looking instead for purely mathematical modern treatment of the problem? I think Carroll's astrophysics text is a good reference in either case, from what I recall. – Cameron L. Williams Dec 01 '14 at 21:37
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    Kepler's second law is (equivalent to) a statement about angular momentum being conserved. As such, it is valid for any central force, of which the inverse-square law is but one example. To proceed further, we need Kepler's first law as well (orbits are ellipses with sun at a focus). See this link for a nice account. – Semiclassical Dec 01 '14 at 22:58
  • @CameronWilliams : I had in mind primarily the math, but the history would also be of interest. – Michael Hardy Dec 01 '14 at 23:48
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    These notes aren't so bad. – Mark McClure Dec 08 '14 at 23:41
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    To state the issue a little differently: Kepler's second law (the constancy of areal velocity) is a differential equation which gives the angular velocity $\dot{\theta(t)}$ as a function of radial position $r(t)$. Hence it isn't reasonable to expect that we can deduce either coordinate $r(t)$ or $\theta(t)$ as functions of time unless we have one additional relationship (e.g. $\ddot{r}\propto -r^{-2}$ (gravitational force law), $r=r(\theta)$ is an ellipse (Kepler's first law).) – Semiclassical Dec 10 '14 at 20:53

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It's a consequence of the rotation invariance of the system. So, for any given $\hat{z}$ axis, perpendicular to the orbit plane, the $z$ angular momentum component is a constant of motion:

$$ M_z=ymv_x - xmv_y =2m\left(\,{1 \over 2}\,r^2 \dot{\theta}\,\right) =2m\,{{\rm d}\text{Area} \over {\rm d}t} $$

Michael Hardy
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Felix Marin
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  • @MichaelHardy You are right. However, the OP didn't ask for a solution. He just said

    $$\sf\mbox{Where are the best expository accounts of the process of solving this equation,... ?}" $$.

     – Felix Marin Dec 09 '14 at 00:41