How do you prove that path of a satellite or a planet is a second degree curve? In other words, how do you prove Kepler's law which states that planets move in elliptical paths?
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asked Mar 5, 2020 at 6:15
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2$\begingroup$ Wikipedia’s article “Kepler orbit” shows how to solve the differential equation of motion to get an ellipse (or a hyperbola, or a parabola, depending on the energy). $\endgroup$– G. SmithCommented Mar 5, 2020 at 6:25
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1 Answer
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The equation of motion is a second order differential equation in time. We could write down the (known) force law and solve it for $(r(t),\theta(t))$. We will use Lagrangian mechanics to proceed. I will highlight the crucial ingredients; the detailed algebra is left as an exercise. See e.g. Goldstein for details.
1) Because the force is central, angular momentum $\vec{L}$ is conserved. This immediately tells us that (a) the motion must be planar(otherwise the direction of $\vec{L}$ changes) and there are only 2 generalized coordinates $r,\theta$, and the angular momentum(conjugate to $\theta$) satisfies $p_\theta=\partial L/\partial\dot{\theta}=l$, a constant. This easily gives us $\theta(t)$.
2) Write the Lagrangian in center of mass coordinates(note that the COM remains fixed if there are no external forces) to simplify things; and note that constants added to Lagrangian mean nothing(so drop them). Write down the Euler-lagrange equations for $r$. Use the above equation involving $\theta$ to eliminate $t$ from the $r$-differential equation(roughly, you will have to convert $dt$ to $d\theta$, for example.)
3) You now have a differential equation in $r,\theta$; solve for $r(\theta)$. This is the locus of coordinates, i.e the orbit. It turns out to be a conic section.
