Understanding Equilibrium Stability in Classical Mechanics: Virtual Work Lecture

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In classical mechanics, the stability of equilibrium can be assessed using the second derivative of potential energy, U. If the second derivative, ∂²U/∂x², is positive, the potential is at a minimum, indicating stable equilibrium. Conversely, if it is negative, the potential is at a maximum, signifying unstable equilibrium. Objects in a potential field tend to move towards lower potential, aligning with the direction of force. Understanding this relationship is crucial for analyzing equilibrium stability in mechanical systems.
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In classical mechanics - virtual work lecture, for determining equilibrium stability we were told that second derivate of potential can be used. How?

I've made a quick google search, but couldn't find anything remarkable.
 
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If \frac{\partial^2U}{\partial x^2} is postive then U is at a minimum (basic calculus - second derivative test). If instead it is negative then U is at a maximum. A point of equilibrium is stable if U is minimum and unstable if U is maximum.
 
Euclid said:
A point of equilibrium is stable if U is minimum and unstable if U is maximum.

I guess this because objects in a potential field tend move through where their potentials get lower?
 
Absolutely! After all, which way does the force point?
 
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