I am self-studying the 3rd edition of Goldstein's Classical Mechanics and I'm having trouble making sense of a certain figure, namely, Figure 3.12 (p. 88), reproduced below.

enter image description here

The figure follows an explanation of the fact that the orbit corresponding to the differential equation for a central force field with potential $V$

\begin{equation*} \frac{\mathrm{d}^{2}u}{\mathrm{d}\theta^{2}} + u = -\frac{m}{\ell^{2}}\frac{\mathrm{d}}{\mathrm{d}u}V\left(\frac{1}{u}\right) \end{equation*}

(in which $u = 1/r$, $\ell$ is the angular momentum, and $m$ is the reduced mass) is symmetric about two adjacent turning points. Therefore, the authors say, the orbit is invariant under reflection about the apsidal vectors. They then point out that we can take advantage of this invariance by noting that, if we know only the portion of the orbit between two turning points, we can construct the rest of the orbit by successive reflections about apsidal vectors "until the rest of the orbit is completed". Apparently the figure is supposed to demonstrate this, but I'm having trouble seeing how.

Evidently, the solid portion represents the known portion of the orbit and the arrows represent the apsidal vectors. The hatched lines then represent the portions of the orbit obtained by reflections. Is the any reason the arrows are pointing in the directions they are, or is it just for sake of generality that they point in different directions?

Also, by saying "until the rest of the orbit is completed," are they implying that the orbit is necessarily closed? It's not immediately clear to me that the orbit obtained by continuing to reflect about the apsidal vectors in the figure would be closed.


1 Answer 1


The orbit ($u$ as a function of $\theta$) apparently has the property of oscillating between two values $u_1$ and $u_2$. Apsidal vectors by definition point to the positions of minimal and maximal $u$. From the drawing we see that those positions are close to, but not exactly 90 degrees apart, so the drawing shows a non-closed orbit.

It depends on the potential $V(u)$ whether this behavior will occur, of course. The solution $u(\theta)$ is either a periodic function, or it goes to a limit, most likely $0$ or $\infty$ in that case, but it could also asymptotically go to some unstable equilibrium value for $u$ which would then of course be precisely at a potential maximum (provided you create a total potential by bringing the $r$ term to the rhs).

And if it is a periodic solution, then the period can be anything (there is nothing in the equation that makes the point $2\pi$ special, you could also see it as some quantity which is a function of time and use $t$ instead of $\theta$). And the periodic solution does not have to be an undistorted sine wave, but we can see that it is time-reversal invariant (or $\theta$- reversal invariant in the original, because there is only an even dependence on $\theta$ throught the second derivative. So you only need to solve one quarter period between one of the minima and a neighboring maximum.

  • $\begingroup$ Thanks for your answer! It makes complete sense. Even in the absence of a closed orbit, is it true that one could carry on reflecting the orbit about the next apsidal vector ad infinitum and continue obtaining more and more of the orbit? $\endgroup$ Feb 26 at 11:37
  • $\begingroup$ Also, if the orbit is not bounded, are the apsidal vectors defined for each period? $\endgroup$ Feb 26 at 11:44
  • 1
    $\begingroup$ Yes, you could have an infinite number of apsidal vectors if the orbit never closes, or just a large number of them if it closes after a lot of revolutions. You could even have only one single vector if it takes exactly one complete revolution to go from maximum to minimum distance (and in the next revolution back up again!) $\endgroup$ Feb 26 at 11:54
  • $\begingroup$ In that last sentence I shoud have said that the apsidal vectors point in just one direction. There are of course still two of them with different length, but there is then only on apsidal $\theta$ value. $\endgroup$ Feb 26 at 12:12

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