Impact on a slab, like a bullet hitting armor or a rock hitting a windshield. If the projectile is long and dense enough to puncture, then it goes through and just keeps going—possibly even accelerating more, if e.g. it’s falling. But if the projectile isn’t past that critical point, it collides and either bounces off or blasts a crater in the surface or slows to a stop or whatever.
When learning a subject, it often takes a lot of time to get to the point where it feels net-positive-value. Before that, each step is another point at which people “bounce off”. But after that, people will just keep delving deeper and deeper.
Similarly, a business might need to invest a lot of effort/money in order to become profitable. Each step before profitability has some chance of the founders/investors giving up. But once past the point of profitability, they’ll likely keep investing more and more.
One thing I notice in the latter two examples is that we don’t necessarily know where the equilibrium point is. We could maybe try to track “how close we are” by looking at e.g. a trendline of a company’s profitability, although we don’t necessarily know how “expensive” the next few steps will be.
I enjoy all these examples. They seem like examples of checkpoints, or milestones. Reconnecting with your examples of balls rolling on a ramp, we might view this as a ball with some initial velocity rolling uphill. If it can make it over the peak, it’ll continue down the other side. Activation energy in a chemical reaction is a related frame.
One way to refine this frame further, and reconnect it with its origin in differential equations, is to consider a system in which the forces endogenous to the system reliably cause us to approach and reach the equilibrium point. In the absence of some external shock, objects will move toward this point, decelerate as they get close, and reliably stop when they arrive. This stands in contrast to a ball rolling uphill, which does not reliably stop when it reaches the peak of the hill, or move toward the peak if it is not pushed.
Nemoto suggested the example of a red light, and I think it’s a good one. We slow to a stop as we approach it, and we’ll stay there as long as the light is red. Once we’re past the red light, though, we move away from it, as we do if some external force shoved us into the intersection when our light’s red and traffic is approaching us on our left or right.
This seems like a case where it’s particularly important to be clear on which forces are internal to the system, and which are external to it. Most real-world cases involve a complex mixture of forces. Semistable equilibria are delicate, since the smallest bump past them results in movement away from the point. In a system facing many disruptions, it may be hard to perceive these semistable equilibrium points, since they’re disguised by these external disruptions.
Yeah, I’m familiar with the meaning in DEs, which is exactly why it seemed weird: the DEs version is the sort of thing which should basically-never happen, because it’s extremely sensitive. Change the parameters even the slightest bit, and we either get two equilibria (one stable, one unstable) or no equilibrium.
Yeah, I think that’s right if we are considering a fast system and are being precise about the zero-velocity point. But the process of deceleration approaching zero at the limit could take a long time, and the forces that risk pushing us to the other side may be rare enough that we can often find objects at or near the equilibrium point.
Impact on a slab, like a bullet hitting armor or a rock hitting a windshield. If the projectile is long and dense enough to puncture, then it goes through and just keeps going—possibly even accelerating more, if e.g. it’s falling. But if the projectile isn’t past that critical point, it collides and either bounces off or blasts a crater in the surface or slows to a stop or whatever.
When learning a subject, it often takes a lot of time to get to the point where it feels net-positive-value. Before that, each step is another point at which people “bounce off”. But after that, people will just keep delving deeper and deeper.
Similarly, a business might need to invest a lot of effort/money in order to become profitable. Each step before profitability has some chance of the founders/investors giving up. But once past the point of profitability, they’ll likely keep investing more and more.
One thing I notice in the latter two examples is that we don’t necessarily know where the equilibrium point is. We could maybe try to track “how close we are” by looking at e.g. a trendline of a company’s profitability, although we don’t necessarily know how “expensive” the next few steps will be.
I enjoy all these examples. They seem like examples of checkpoints, or milestones. Reconnecting with your examples of balls rolling on a ramp, we might view this as a ball with some initial velocity rolling uphill. If it can make it over the peak, it’ll continue down the other side. Activation energy in a chemical reaction is a related frame.
One way to refine this frame further, and reconnect it with its origin in differential equations, is to consider a system in which the forces endogenous to the system reliably cause us to approach and reach the equilibrium point. In the absence of some external shock, objects will move toward this point, decelerate as they get close, and reliably stop when they arrive. This stands in contrast to a ball rolling uphill, which does not reliably stop when it reaches the peak of the hill, or move toward the peak if it is not pushed.
Nemoto suggested the example of a red light, and I think it’s a good one. We slow to a stop as we approach it, and we’ll stay there as long as the light is red. Once we’re past the red light, though, we move away from it, as we do if some external force shoved us into the intersection when our light’s red and traffic is approaching us on our left or right.
This seems like a case where it’s particularly important to be clear on which forces are internal to the system, and which are external to it. Most real-world cases involve a complex mixture of forces. Semistable equilibria are delicate, since the smallest bump past them results in movement away from the point. In a system facing many disruptions, it may be hard to perceive these semistable equilibrium points, since they’re disguised by these external disruptions.
Yeah, I’m familiar with the meaning in DEs, which is exactly why it seemed weird: the DEs version is the sort of thing which should basically-never happen, because it’s extremely sensitive. Change the parameters even the slightest bit, and we either get two equilibria (one stable, one unstable) or no equilibrium.
Yeah, I think that’s right if we are considering a fast system and are being precise about the zero-velocity point. But the process of deceleration approaching zero at the limit could take a long time, and the forces that risk pushing us to the other side may be rare enough that we can often find objects at or near the equilibrium point.