Physical meaning of stable equilibrium

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jonathan.15
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What is the physical meaning of an object to be in stable equilibrium?

Would it make sense to say that the object wants to stay at that position and any movement will cause the restoring force to move the object back to equilibrium?
 
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Yes. That is correct.

Also, if the object is being acted on by a conservative force then the same condition means that it is at a local minimum of the potential energy.