# A doubt in a Wikipedia article discussing Bertrand's Theorem in Central force motion

Wikipedia on Bertrand's theorem, when discussing the deviations from a circular orbit says:

...The next step is to consider the equation for $$u$$ under small perturbations $$\eta \equiv u-u_{0}}$$ from perfectly circular orbits.

(Here $$u$$ is related to the radial distance as $$u=1/r$$ and $$u_0$$ corresponds to the radius of a circular orbit ) ...

The deviations are as

The solutions are $$\eta (\theta )=h_{1}\cos(\beta \theta ),}$$

For the orbits to be closed, $$β$$ must be a rational number. What's more, it must be the same rational number for all radii, since β cannot change continuously; the rational numbers are totally disconnected from one another

Why does $$\beta$$ have to be the same rational number for all radii at which a circular orbit is possible?

I understand why it should be rational, but why the same number for all radii?

The variable $$\beta$$ must vary continuously with the radius because it is defined in terms of another function $$J$$ that varies continuously with the radius. Now, suppose there are radii $$r_1$$ and $$r_2$$ such that $$\beta(r_1)=3$$ and $$\beta(r_2)=3.2$$. Because $$\beta$$ is continuous, there must be a radius $$r_3$$ between $$r_1$$ and $$r_2$$ such that $$\beta(r_3)=\pi$$. This cannot happen because $$\beta$$ must be rational, and there is no such thing as varying continuously over the rational numbers. So, $$\beta$$ cannot vary and must be constant.
• But β is defined only at specific radii where we've a circular orbit. So β isn't a continuous function to begin with as is defined in the article : $\beta ^{2}\equiv 1-J'(u_{0})$ Oct 27, 2021 at 17:08
• @Kashmiri Circular orbits can occur at any radius, so $u_0$ is a continuous variable. If you can calculate different $\beta$s at different radii, then $\beta$ must vary continuously. Oct 27, 2021 at 17:19