Hooke's law, Bertrand's theorem and closed orbits

In summary, Bertrand's Theorem states that the only forces that result in closed orbits are Hooke's law and the attractive inverse square force. The conversation discusses trying to prove this for Hooke's law and ultimately concludes that it results in three simple harmonic motions, resulting in a closed orbit. This can be seen in the illustration on a pound note, where the sun is placed in the center of an elliptical orbit, much to Newton's dismay.
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Kashmir
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Bertrand's Theorem says : the only forces whose bounded orbits imply closed orbits are the Hooke's law and the attractive inverse square force.

I'm looking at the hookes law ##f=-k r## and try to see explicitly that the orbit is indeed closed.

I use the orbit equation ##\frac{d^{2} u}{d \theta^{2}}+u=\frac{-m}{l^{2} u^{2}} f\left(\frac{1}{u}\right)## with the force given as ##f=-k r## ,therefore I get ##\frac{d^{2} u}{d \theta^{2}}+u=+\frac{mk}{l^{2} u 3}## as the equation defining the trajectory.

However neither can I solve this nor can I see that the equation implies a closed orbit.

Can you please help me.
 
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  • #2
Kashmir said:
I'm looking at the hookes law ##f=-k r## and try to see explicitly that the orbit is indeed closed.

Hooke's law gives you harmonic oscillation along each axis in the orbital plane, with a period independent of max. amplitude, thus the same for both axes and equal to the orbital period:
https://en.wikipedia.org/wiki/Harmonic_oscillator
 
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  • #3
A.T. said:
Hooke's law gives you harmonic oscillation along each axis in the orbital plane, with a period independent of max. amplitude, thus the same for both axes and equal to the orbital period:
https://en.wikipedia.org/wiki/Harmonic_oscillator
Thank you. I got it. We've three SHM along three axis which are periodic, hence a closed orbit.
 
  • #4
Kashmir said:
We've three SHM along three axis which are periodic, hence a closed orbit.

You can see the SHM orbit on the pound note below, where they placed the sun in the middle of an elliptical orbit. Newton doesn't look too happy about this.

One%2BPound%2BNote%2Bwith%2BIsaac%2BNewton.jpg
 
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1. What is Hooke's Law?

Hooke's Law is a principle in physics that describes the relationship between the force applied to a spring or elastic material and the resulting displacement. It states that the force applied is directly proportional to the amount of displacement, as long as the elastic limit of the material is not exceeded.

2. Who discovered Hooke's Law?

Hooke's Law was discovered by English scientist Robert Hooke in the 17th century. He first described this relationship in his book "Micrographia" in 1660.

3. What is Bertrand's Theorem?

Bertrand's Theorem is a mathematical theorem that states that in a central force field, there are only three types of stable closed orbits: circular, radial (straight line), and planar elliptical orbits. This theorem was first proven by French mathematician Joseph Bertrand in 1873.

4. How does Bertrand's Theorem relate to Hooke's Law?

Hooke's Law can be applied to the study of closed orbits in a central force field, as the force acting on an object in these orbits is directly proportional to its displacement from the center. This relationship is similar to the relationship described in Hooke's Law.

5. What are closed orbits?

Closed orbits are a type of orbit in which an object moves around a central point in a repeating pattern without ever escaping or colliding with the central point. This can occur in a variety of systems, such as planetary orbits around a star or electrons orbiting an atomic nucleus.

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