# Physical interpretation of the "total potential energy"

• vco
In summary, the principle of minimum total potential energy is a way of deriving equilibrium equations for an elastic body under conservative forces. It states that out of all the possible displacement fields that satisfy the boundary conditions, the one which minimizes the total potential energy satisfies equilibrium. This method is commonly used in textbooks, but I have been wondering for years what exactly is the physical interpretation of the "potential energy of the external forces", and consequently, the physical interpretation of the total potential energy itself. It is also worth noting that the lost potential energy is turned into heat in a quasi-static case.
vco
The principle of minimum total potential energy is frequently used in solid mechanics as an elegant way of deriving the equilibrium equations for an elastic body under conservative forces. This method states that out of all the possible displacement fields that fulfill the boundary conditions, the one which minimizes the total potential energy satisfies equilibrium.

In textbooks, the total potential energy is defined as the sum of the elastic energy of the body and the potential energy of the external forces. The elastic energy is a well-defined concept, but I have been wondering for years what exactly is the physical interpretation of the "potential energy of the external forces", and consequently, the physical interpretation of the total potential energy itself.

A simple example of the application of the method: for a linear spring with stiffness ##k## loaded with a force ##F##, the elastic energy is $$U=\frac{1}{2}ku^2,$$ and the potential energy of the external force is defined as $$V=-Fu,$$ where ##u## is the displacement of the spring. The total potential energy of the system is $$\Pi=U+V=\frac{1}{2}ku^2-Fu,$$and by minimizing ##\Pi## with respect to ##u## we end up with the obvious equilibrium equation $$ku-F=0.$$ In every textbook I have seen, the total potential energy and the associated principle are introduced as if they were some fundamental physical concepts that need no explanation or derivation. Even the name "total potential energy" sounds confusing since it cannot be the energy stored in the system; the stored energy is the elastic energy, is it not?

Also, the Wikipedia article says something about the lost potential energy being turned into heat, how does this fit in? Is this related to the forces being applied instantaneously vs quasi-statically?

So, could someone please explain what is the physical meaning of the
1. potential energy of the external forces?
2. total potential energy?

Newton's second law, ## \vec F=m \ddot{\vec x} ## gives you the acceleration of a particle at any moment in time. The particle is in Equilibrium if there is zero net force acting on it. Now if we define ## \vec F=-\vec \nabla V ##, the equilibrium point is given by ## \vec \nabla V=0 ## which gives the extremums of the potential. And also because its force that is observable and two potentials differing only in an additive constant give the same force, the total potential is not a physical quantity and only difference in potential is physical.

vanhees71
vco said:
Even the name "total potential energy" sounds confusing since it cannot be the energy stored in the system; the stored energy is the elastic energy, is it not?
Not all of the system's energy is elastic. Some of the system's energy is stored in F.

Dale said:
Not all of the system's energy is elastic. Some of the system's energy is stored in F.
But if we were to suddenly make ##F=0##, would not the spring start to vibrate with a total energy equal to the elastic energy of the original scenario?

vco said:
But if we were to suddenly make ##F=0##,
Then the potential for F would be time dependent and therefore energy would not be conserved.

vanhees71
I am still confused.

Another thing to increase my confusion:
Since the potential energy of the external forces is by definition the negative of the work they perform, there seems to be a "hidden" assumption that the external forces are applied instantaneously and not quasi-statically.

In a quasi-static case, the work performed by the external forces is equal to the elastic energy. In the instantaneous case which the principle suggests, the work performed by the external forces is greater than the elastic energy.

In the example, the spring begins to vibrate due to the external force attaining its final value immediately. The external force must attain its final value immediately since the work it performs (the negative of its potential energy) is defined to be ##Fu##. The vibration seems to be assumed to dissipate due to damping present in the system, and finally the equilibrium position is attained.

Where does this reasoning go wrong, or does it?

vco said:
Since the potential energy of the external forces is by definition the negative of the work they perform, there seems to be a "hidden" assumption that the external forces are applied instantaneously and not quasi-statically.
I have no idea what you mean by this.

vco said:
In the example, the spring begins to vibrate due to the external force attaining its final value immediately
This is incorrect. The vibration will happen with or without the external force, depending on the initial conditions.

Oscillations happen when you have a potential with a minimum. Adding a linear potential may change the location of the minimum, but doesn't determine whether or not there is a minimum

vco said:
The vibration seems to be assumed to dissipate due to damping present in the system, and finally the equilibrium position is attained.
Sorry, I had missed this earlier and had forgotten the original context regarding he principle of minimum potential energy.

Yes, you are correct. Vibrations are assumed to have already dissipated. Alternatively, you can consider that you are looking for equilibrium solution which is, by definition, the solution which is not a function of time.

vco said:
Since the potential energy of the external forces is by definition the negative of the work they perform, there seems to be a "hidden" assumption that the external forces are applied instantaneously and not quasi-statically.

Dale said:
I have no idea what you mean by this.

Imagine the spring is from the suspension of a car, and you compress it in a press. Or that the spring is one of those demo springs found in physics classrooms, and you compress it with your hand. You do not apply the force instantly as ##F##, but you increase it from zero to ##F##. The work the force performs is not ##Fu## since the force is not constant during the deformation. If you apply the force infinitely slowly from zero to ##F##, i.e., quasi-statically, then the work performed by the force is equal to the elastic energy ##\frac{1}{2}ku^2##, which is half of ##Fu##. In the instantaneous case, the difference between the work performed by the external force and the elastic energy of the spring in equilibrium manifests itself as vibration, which then dissipates.

The principle of minimum total potential energy implies that the force is applied instantaneously: the potential energy of the external force is stated as ##-Fu## and hence the work it performs is ##Fu##, which in turn implies that the force is constant during the deformation.

Someone else has also wondered this issue: http://physics.stackexchange.com/qu...rce-on-particle-attached-to-spring-is-equal-t

Now, I know that the principle of minimum total potential energy applies only to statics, not dynamics. And I also know that the spring example introduced in my first post is static by definition. But when you move from the idealized world to the real world, the example is no longer strictly static. But you still get the correct solution (the equilibrium equation) by applying the principle, even if the work performed by the external force is different than in the idealized world.

I must admit that I am an engineer, so I might be looking at things from a different perspective.

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vco said:
Imagine the spring is from the suspension of a car, and you compress it in a press. Or that the spring is one of those demo springs found in physics classrooms, and you compress it with your hand. You do not apply the force instantly as ##F##, but you increase it from zero to ##F##. The work the force performs is not ##Fu## since the force is not constant during the deformation. If you apply the force infinitely slowly from zero to ##F##, i.e., quasi-statically, then the work performed by the force is equal to the elastic energy ##\frac{1}{2}ku^2##, which is half of ##Fu##. In the instantaneous case, the difference between the work performed by the external force and the elastic energy of the spring in equilibrium manifests itself as vibration, which then dissipates.
You are way overthinking this. We are looking at static solutions, so there is no "apply". The force simply is, it has always been there and will always be there. That is what it means to be static, it doesn't change over time. There is no distinction between quasi-static or instantaneous, since either one is a force changing over time, and we are looking only for static solutions so both the instantaneous and the gradual application are explicitly excluded by our assumption.

Yes, this is unrealistic, but that is not the point. In science and physics and especially in engineering we make assumptions that simplify things. We know that our assumptions are wrong at some level, but they can often serve as a useful approximation. Here the assumption that we are making is the static assumption, so nothing will depend on time.

vco said:
Now, I know that the principle of minimum total potential energy applies only to statics, not dynamics. And I also know that the spring example introduced in my first post is static by definition. But when you move from the idealized world to the real world, the example is no longer strictly static. But you still get the correct solution (the equilibrium equation) by applying the principle, even if the work performed by the external force is different than in the idealized world.
Yes, this is why we teach and use this particular approximation. The reason it works is that damping forces are a function of the velocity rather than the coordinate, so they don't affect the asymptotic position. So the asymptotic position of a damped system is the same as the static position of the undamped system.

vco said:
I must admit that I am an engineer, so I might be looking at things from a different perspective.
I am an engineer too. You should get used to making assumptions and approximations. You need to have your assumptions in mind when you analyze a problem for two reasons, first they are important because they make analysis easier, second they are important because they can lead to big errors when they are violated.

Thank you. That was an excellent post.

I am used to making assumptions and approximations. It is just that if there is even the slightest sense of doubt in my mind regarding the mathematical and logical consistency of the "foundations", I feel uncomfortable. It is not satisfactory to know perfectly how to use something and know when it can and cannot be applied, but not knowing why it works at the fundamental level.

I gave things some thought and came to the conclusion that the reason for the confusion was that I did not fully understand the concept of a potential energy function in general.

Ever since from high school I had always associated potential energy functions of forces to either internal forces (elasticity) or external force fields (gravitation). But of course also "regular" external forces not arising from force fields (contact forces) have potential energy as long as they are conservative, even though this is less intuitive.

In the idealized world, a conservative external force would continue to perform work to the body from here to eternity should all other forces dissappear, just like the gravitational force in an infinite gravitational field would. We have defined the external force to be there, and there it shall be no matter what happens. A constant gravitational force is no different from a constant conservative external force in this context.

I think the following misleading convention has contributed to my confusion: in textbooks, the gravitational potential energy is always included in the total potential energy, but the potential energy of the external force is usually excluded from the total potential energy in textbooks not specifically dealing with applied/analytical/structural mechanics.

So unless you disagree, I consider myself enlightened :)

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Dale said:
Not all of the system's energy is elastic. Some of the system's energy is stored in F.
What do you mean by 'stored n F' f it is not potential energy

Dale said:
Not all of the system's energy is elastic. Some of the system's energy is stored in F.

lychette said:
What do you mean by 'stored n F' f it is not potential energy

I think he means that if we wanted to bring the system back from the loaded position to the original position while keeping ##F## as it is, then the work required would be equal to the change in the total potential energy. In this process, the value of the total potential energy would increase from ##\frac{1}{2}ku^2 - Fu = -\frac{1}{2}ku^2## back to zero.

It is a bit tricky to understand if you consider it from a too practical aspect. Of course, the energy that would be released by the spring if ##F## was suddenly removed would be the elastic energy (which in equilibrium appears to be, at least in this example, the negative of the total potential energy). But as long as ##F## is there, we have to include its potential energy in the total potential energy of the system.

Imagine that ##F## was caused by gravity. In this case, it is quite clear that part of the energy of the system is stored in ##F##. Think about it, and see that it makes no difference whether ##F## is an externally applied conservative force or a gravitational force. We cannot turn off gravity, and we cannot turn off an external force since we have defined it to be there.

This is how I see it. However, note that I figured this out quite recently, so I am not that sure whether this is correct.

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Dale
lychette said:
What do you mean by 'stored n F' f it is not potential energy
It is potential energy, just not elastic potential energy. The OP didn't specify the nature of F, but whatever its nature it clearly has an associated potential energy.

That potential energy is the same as the gravitational potential energy of an object with mass ##F/g##, so barring any other information that is what I would assume.

Dale said:
It is potential energy, just not elastic potential energy. The OP didn't specify the nature of F, but whatever its nature it clearly has an associated potential energy.

That potential energy is the same as the gravitational potential energy of an object with mass ##F/g##, so barring any other information that is what I would assume.
Many apologies for my typing error...
'...stored in F' if it is not potential energy.

Dale said:
The OP didn't specify the nature of F, but whatever its nature it clearly has an associated potential energy.

The nature of the force in the example is simply that it is constant in magnitude and direction. Therefore, it is conservative and has a potential energy function, like you said. No other restrictions.

Dale
I'm confused on this also. Let's say you impose work (energy) with magnitude of F*u, but the strain energy magnitude is less than the work. Where did the extra energy go? I expected that the imposed work (F*u) would all go into strain energy.

bcl said:
Lets say you impose work (energy) with magnitude of F*u, but the strain energy magnitude is less than the work. Where did the extra energy go?
Either to another potential energy or to kinetic energy.

The way I look at this is to start by imagining a simple experiment. Imagine a spring suspended vertically with a weight hanger at the lower end. Adding a weight to the spring results in a final equilibrium extension but the way that the system reaches this depends on the method by which the weight is added. There are two extreme types of method:

1. The hanger is held in place as the weight is added and then quickly released. This will result in the weight overshooting the equilibrium extension and this will set the system into damped vibrations with energy being lost as heat. Eventually the system will come to rest at the equilibrium extension

2. The hanger and weight are not released quickly but support is given to the weight so that it moves slowly to its final extension. In this case work done to move the system slowly will be equal to the energy lost as heat in the above case.

It turns out that whatever method is used the energy converted in reaching the equilibrium extension is greater than the final elastic energy stored in the spring.

vco said:
This method states that out of all the possible displacement fields that fulfill the boundary conditions, the one which minimizes the total potential energy
it is better to say about extremums
and all this theory works provided all the forces are potential. Roughly speaking it works in autonomous Hamiltonian systems

## 1. What is the definition of total potential energy?

The total potential energy is a concept in physics that refers to the sum of all the potential energies of a system. It includes gravitational potential energy, elastic potential energy, and any other form of potential energy present in the system.

## 2. How is total potential energy calculated?

Total potential energy is calculated by adding together all the potential energies present in a system. This can be done by using the appropriate equations for each type of potential energy, such as the equation for gravitational potential energy or the equation for elastic potential energy.

## 3. What is the significance of understanding total potential energy?

Understanding total potential energy is important because it allows us to predict and analyze the behavior of physical systems. It helps us understand how energy is stored in various forms and how it can be transformed into different types of energy.

## 4. How does total potential energy relate to the concept of conservation of energy?

The concept of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. Total potential energy is a part of this overall energy conservation principle, as it represents the potential for energy to be transformed into other forms.

## 5. Can total potential energy be negative?

Yes, total potential energy can be negative. This typically occurs when there is a negative potential energy present in the system, such as in the case of a ball rolling down a hill. The negative value simply indicates that the system has the potential to release energy and do work.

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