I Why must the oscillatory period of a stable orbit be constant at all distances

[Happiness]

Consider a stable circular orbit (with a central force) subjected to small perturbations. The orbit equation is given by (3.45).

The text argues that the $\beta$ in (3.46) must be a constant over the domain of $r_0$: "Otherwise, since $\beta$ can take on only discrete rational values (for closed orbits), the number of oscillatory periods would change discontinuously with $r_0$, and indeed the orbits could not be closed at the discontinuity." (5th last line in the last paragraph attached below)

I don't understand this argument. I could have $\beta=4$ when $r=r_0=1$m and $\beta=5$ when $r=r_0'=2$m. The orbits at these values of $r$ are closed. And I can have the orbit not to be closed at other values of $r$, which I supposed are the discontinuities referred to by the text. "Indeed the orbits could not be closed at the discontinuity." But so what? We only demand the orbit to be closed at distances $r$ where the orbit is stable and circular.

 

[mfb]

You would need a really weird central force to have multiple regions of stability and instability, but it is possible to write down such a force.

 

[Happiness]

You would need a really weird central force to have multiple regions of stability and instability, but it is possible to write down such a force.

Stable circular orbits occur at distances $r$ where $V'$ is a local minimum. Local minima are separated from one another (examples attached below), unless $V'$ is a horizontal line in the neighborhood. In fact, the examples given only have one minimum or maximum. So I supposed the domain of $r_0$ must be discontinuous. That's why I can't make sense of the argument used by the book.

 

[mfb]

They only have one minimum or maximum for given E and L. Different orbits have different E and L.