Is Stable Equilibrium Possible Without Potential Energy Minima?

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The discussion centers on the Lejeune Dirichlet theorem, which states that stable equilibrium occurs when potential energy has minima, but seeks examples of stable equilibrium without such minima. The Lagrangian points L4 and L5 are mentioned as examples of stable points in a dynamic, rotating system. It is noted that the theorem primarily applies to small oscillations, which are not present in this context. The rotation of planets around the sun is suggested as a simpler example of dynamic equilibrium. Overall, the conversation explores the nuances of stability in equilibrium beyond the traditional understanding of potential energy minima.
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Lejeune Dirichlet theorem says that when potential energy has minima then equilibrium is stable, but that is sufficient condition. Can you give me example or examples where potential energy hasn't minima and equilibrium is stable. Tnx
 
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Ok. But Lejeune Dirichlet theorem is for small oscilation. I don't see any oscillation in here?
 
These are stable points, but only in a dynamic, rotating system. Stable implies that you can have small oscillations around the point of equilibrium.

Actually, the rotation of planets around the sun would be a simpler example of a dynamic equilibrium.
 

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