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

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Bertrand's Theorem states that only forces leading to closed orbits are those described by Hooke's law and the attractive inverse square law. The discussion focuses on demonstrating that Hooke's law, represented by the force equation f = -k r, results in closed orbits through the derived trajectory equation. The analysis reveals that the motion can be understood as three simple harmonic motions (SHM) along different axes, which are periodic and thus yield a closed orbit. The periodic nature of these oscillations is independent of amplitude, reinforcing the closed orbit concept. The conversation concludes with a reference to a visual representation of SHM orbits, emphasizing their elliptical nature.
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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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
 
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.
 
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
 
Hello everyone, Consider the problem in which a car is told to travel at 30 km/h for L kilometers and then at 60 km/h for another L kilometers. Next, you are asked to determine the average speed. My question is: although we know that the average speed in this case is the harmonic mean of the two speeds, is it also possible to state that the average speed over this 2L-kilometer stretch can be obtained as a weighted average of the two speeds? Best regards, DaTario
This has been discussed many times on PF, and will likely come up again, so the video might come handy. Previous threads: https://www.physicsforums.com/threads/is-a-treadmill-incline-just-a-marketing-gimmick.937725/ https://www.physicsforums.com/threads/work-done-running-on-an-inclined-treadmill.927825/ https://www.physicsforums.com/threads/how-do-we-calculate-the-energy-we-used-to-do-something.1052162/
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