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## Stability of equilibria

Equilibrium points can be thought of as follows. If an object is placed at rest at an equilibrium point, it'll stay there for all time. This is therefore a point where the force acting on the object is zero. This therefore corresponds to a point where the slope of the potential energy curve is zero.

Now let's discuss shapes of various potential energies in one dimension, and see how this leads to ideas of stability.

First we'll consider this nice concave shaped potential function pictured below

Below it is sketched the force, obtained by taking the negative derivative of the potential energy. On top is pictured the force a red ball will feel when at various x positions. If the ball is to the left of the minimum, the force is positive and therefore points to the right. Conversely, if the ball is to the right of the potential minimum, it'll feel a force to the left. It feels no force at the minimum because there the derivative is zero.

So if the ball starts off at rest at the minimum, it'll stay there. If it starts at rest away from it, it's pulled towards the minimum. Suppose it starts out like the red ball on the right. It'll start moving towards the minimum, transferring potential energy to kinetic energy. Since energy is conserved, the maximum potential energy is when the kinetic energy is zero. That energy , is shown by the horizontal dashed line. So this means that the particle can't move up to a higher starting point than this horizontal line. This means that the motion is bounded. We call this minimum, for obvious reasons, a stable equilibrium point.

Now let's look at the upside-down version of the same potential

Now everything is reversed. The force is the negative of what it was before. So if a red ball starts on the left of the potential maximum, it'll be pushed away from it. If it starts on the right it'll also be pushed away from the maximum. It feels no force at the maximum itself so if a ball starts out at rest at the maximum, it'll stay there for all time.

But what we can see is that if it starts out at rest just slightly away from the maximum, then it'll continue to move away. In this case the motion is unbounded. We call this maximum, for equally obvious reasons, an unstable equilibrium point.

Lastly consider the potential shown below

Here the force is zero over a finite range in x. If a particle starts off at rest any point on the top of this curve, it'll remain there. It's neither pulled one way or the other. Such a situation is called a neutral equilibrium.

Intuitively, it seems that the potential energy is pretty easy to understand. Take the case of the first curve we discussed, that of a stable equilibrium. You could think of the potential energy as being a salad bowl, and the red balls as cherry tomatoes (whole and organically grown) that are placed, with a little lubricating dressing, at various starting positions inside the bowl. They'll always roll down towards the center of the bowl. See, the person that invented the salad bowl had to have understood something about stable equilibria. If instead the inventor had got confused about minus signs, they may have mis-invented an upside-down salad bowl. This would not be a hit with guests at a dinner party. The guests would have difficulty helping themselves to those tomatoes, seeing them instead careening across the table, spreading your vinagrette all over that beautiful white tablecloth you'd purchased at Kmart and ending up eventually on someone's lap. This is the case of an unstable equilibrium, also pictured above.

The case of a neutral equilibrium is what I like to use at my place, use no bowl at all, you just place things directly on the table. This saves on washing up.

Next: Non-conservation of mechanical energy Up: One dimensional potential energy Previous: solution
Josh Deutsch 2003-02-02