# Exploring Bertrand's Theorem: Why is β the Same Rational Number for All Radii?

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• Kashmir
In summary: Therefore, if β is irrational, for example, you would get a non-closed orbit. In summary, Bertrand's theorem states that for a closed orbit to be possible at all radii, β must be the same rational number.
Kashmir
Wikipedia on Bertrands theorem, when discussing the deviations from a circular orbit says:
>..."The next step is to consider the equation for ##u## under small perturbations ##{\displaystyle \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 solutions are
##{\displaystyle \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?

Kashmir said:
"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 ?
As the quote states, rational numbers are totally disconnected from one another, meaning that between two rational numbers you have non-rational numbers. So the rational number β cannot change continuously, while you continuously vary the radius.

Kashmir

## 1. What is Bertrand's Theorem?

Bertrand's Theorem is a mathematical theorem named after French mathematician Joseph Bertrand. It states that for any circle, there exists at least one chord that is the same length as the radius of the circle.

## 2. Why is Bertrand's Theorem important?

Bertrand's Theorem has many applications in geometry and statistics. It is used to prove other theorems and has implications in probability and combinatorics. It also has practical applications in designing bridges and other structures.

## 3. What is β in Bertrand's Theorem?

β is the angle formed by the two radii of a circle and the chord that is the same length as the radius. It is also known as the central angle.

## 4. How is β calculated in Bertrand's Theorem?

β can be calculated using the formula β = 2arcsin(r/c), where r is the radius of the circle and c is the length of the chord that is the same length as the radius.

## 5. Why is β the same rational number for all radii in Bertrand's Theorem?

This is because the ratio of the length of the chord to the radius is always the same, regardless of the size of the circle. This ratio is equal to sin(β/2), which is a rational number. Therefore, β must also be a rational number.

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