Clarification in the definition of stable and unstable equilibrium

I am studying about types of equilibrium which are stable, unstable and neutral.
In the definitions of these equilibriums they used the term "position of equilibrium". Like in stable equilibrium, after a slight displacement from the position of equilibrium, the body has the tendency to return to its position of equilibrium unlike that of unstable equilibrium.

But I am confused as to what does the "position of equilibrium" represents? I mean, in unstable equilibrium too, the body comes ultimately to equilibrium when it comes to rest. Then how to stable equilibrium differs with unstable one?

Any clarification will be highly appreciated.

• second derivative of potential energy is an indicator of stable or unstable equilibrium for one dimensional motion ,you can calculate it by checking the sign of the second derivative. if it is positive then you can say it is a stable equilibrium and if it is negative then you say it is an unstable equilibrium. Commented Jun 17, 2021 at 7:05
• in unstable equilibrium, the body does come to equilibrium but this is a different equilibrium that happens to be a stable equilibrium. Commented Jun 17, 2021 at 7:14
• in unstable equilibrium too, the body comes ultimately to equilibrium when it comes to rest. can u elaborate? is there a mental picture you have? Commented Jun 17, 2021 at 7:59

Equilibrium is a property of the state of a system. In particular, equilibrium as to do with the energy (alternatively entropy) of the system.

"Position of Equilibrium" simply refers to various portions of an energy vs state graph of a system. Here is a quick hand drawn graph:

Notice that all equilibrium points are characterized by 0 slope (first derivative). Within that the types of equilibrium is characterized by its extrema (second derivative):
i. Stable if second derivative is negative
ii. Neutral if second derivative is zero
iii. Unstable if second derivative is positive.

PHYSICALLY,

1. Stable Equilibrium - is when the systems energy is the lowest (locally). Now its a central principle in physics that any system tries to minimize its energy (equivalent to saying that any system tries to maximizes its entropy). Hence, stable equilibrium is the most sort after equilibrium position of a system and a lot lot of theory has been developed in this region. This equilibrium is characterized by the fact that "any slight perturbation will tend to make the system oscillate about this point" until eventually the extra energy gets dissipated and the system settles back to the stable state.

2. Unstable Equilibrium also called a metastate in literature, the reason being its highly unstable state and any slight perturbation (even thermal fluctuation) can be enough to drastically change the state.

3. Neutral Equilibrium is a directional equilibrium in the sense that, towards one side it is stable, and unstable on the other.

To illustrate using @BillWatts example (posted as an answer to the OP). Here the height of the bead from ground represents energy (potential energy), and states are represented by position of bead on the x-axis.

An example. Consider a hemispherical bowl and a marble. Put a marble in a bowl and the position of equilibrium is the bottom of the bowl where the potential energy is a minimum. Any small displacement from this position will force the marble to return to its position of equilibrium. This is a stable equilibrium.

Now, turn the bowl over so we have the shape of a dome. The position of equilibrium will be the top of the bowl where the potential energy is a maximum. This will be an unstable equilibrium since any displacement will cause the marble to fall further from the equilibrium position and on its own will never return.

Position of equilibrium has nothing to do with nature of equilibrium. Its defined as the point of $$0$$ net force and torque on the test particle.

You already seem to know what is stable equilibrium. The important concept there is that displacement induced restorative forces try to bring a system back to its original location.

In unstable equilibrium, the forces tend to take the system away from its initial equilibrium point. Such systems usually don't go back to their initial state. Even if they did$$^1$$ it wouldn't make that point stable. Its the diverging nature of the induced restorative forces in the neighborhood of that pt. and not the coincidence of having returned to it that makes the point unstable. On the contrary, in stable equilibrium, return to the initial point for small displacements, is a necessity.

$$^1$$ if they have excess energy initially

• That means position of equilibrium here means the point where the body is resting before the displacement . It does not refer to the state of equilibrium which is when the body is in equilibrium ?
– Esha
Commented Jun 17, 2021 at 12:40
• @Esha if the body is in equilibrium, stable unstable or saddle, its pos. before displacemnet is the "pos. of equilib." state of equilib. is whther its stable, unstable or saddle Commented Jun 17, 2021 at 15:37