Is Stable Equilibrium Possible Without Potential Energy Minima?

[matematikuvol]

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

 

[M Quack]

The Lagrangian points L4 and L5.

http://en.wikipedia.org/wiki/Lagrangian_point

 

[matematikuvol]

Ok. But Lejeune Dirichlet theorem is for small oscilation. I don't see any oscillation in here?

 

[M Quack]

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.

 

[PJC01]

Yes, there are several examples where potential energy does not have a minimum and yet the equilibrium is stable. One example is a simple pendulum. The potential energy of a pendulum does not have a minimum, but the equilibrium position is stable because any displacement from the equilibrium position will result in a restoring force that brings the pendulum back to its original position. Another example is a ball resting on top of a hill. The potential energy of the ball does not have a minimum, but the equilibrium is stable because any small displacement from the top of the hill will result in a gravitational force that brings the ball back to the top. In both of these examples, the equilibrium is stable due to the presence of a restoring force that counteracts any external forces acting on the system. Therefore, the Lejeune Dirichlet theorem is not a necessary condition for stable equilibrium.