# Is conservation of energy derived from the work energy theorem?

• etotheipi
In summary, the two distinct approaches to energy problems used by different authors are variations on the work energy theorem and defining a system boundary and setting the change in total energy of that system to the external work done on all components. The work energy theorem involves summing work energy equations for each particle in the system and accounting for overlapping terms with potential energy. This results in the equation Wext = ΔT + ΔU + ΔEth = ΔE, which is commonly used for conservation of energy. However, this approach does not account for heat transfer and may not be as fundamental as other laws such as the first law of thermodynamics. On the other hand, defining a system and using the work energy equation for all forces and a conservative potential allows
etotheipi said:
Ah, okay. I suppose my original question was a little misleading. Apologies!
Ok, so now I am thoroughly confused. What is your intention/goal? In particular regarding the “fundamental-ness” of the work energy theorem and the conservation of energy.

Dale said:
Ok, so now I am thoroughly confused. What is your intention? In particular regarding the “fundamental-ness” of the work energy theorem.

It's more because I'm working from Kleppner and Kolenkow, and they gave a proof of the conservation of mechanical energy via the work energy theorem. And quite a few introductory mechanics textbooks sidestep into conservation of energy in the same way, after first integrating Newton's second law.

This seemed a little odd, not least since you mentioned that there are other pathways for transfer of energy within a system. And to make matters worse, different authors all treat energy slightly differently. Morin/Halliday/Sherwood all make extensive reference to the concept of defining systems and treating energy transfers across the boundary, whilst others will derive most energy relations directly from the work energy theorem.

So really I'm finding it's just a little bit of a chicken and the egg situation. I think now I've come to the conclusion that work-energy and the first law of thermodynamics approaches are distinct, though similar. And one can derive special cases of one from the other.

I do apologise for being daft, and many thanks for your help on this!

vanhees71 and Dale
etotheipi said:
It's more because I'm working from Kleppner and Kolenkow, and they gave a proof of the conservation of mechanical energy via the work energy theorem. And quite a few introductory mechanics textbooks sidestep into conservation of energy in the same way, after first integrating Newton's second law.
The conservation of energy is far more general than the work energy theorem, so I am skeptical that such a derivation is valid. Either you must add something beyond the work energy theorem itself to make the derivation more general or you must limit the derivation to a restricted set of situations in order to apply the theorem.

However, as a pedagogical tool I could see the value. Generally net forces are introduced before energy is introduced, so you can point out conservation of energy as being implied by things they already know. In the pedagogical case the fact that it is not a general derivation is fine, it is just a motivational example and does not need to apply generally.

vanhees71 and etotheipi
Just to close off the thread, I'll summarise in light of @jbriggs444's post.

The total work done on any system equals the change in kinetic energy of that system.

However, for particles and rigid bodies the total work equals the external work (there is no internal work), whilst for deformable bodies the total work equals the external plus the internal work.

So we can only write "The total external work done on a particle/rigid body equals the change in kinetic energy of that body".

This is usually what we're interested in when we do a mechanics problem.

conservation of energy is true, you can simply consider the ideal gas law
https://en.wikipedia.org/wiki/Ideal_gas_law

if you consider adiabatic compression (i.e. all the energy is contained within the gas even if it is compressed)

the equations are like :
p1 . v1 = p2 . v2.
so in compression you can have v2 being smaller than v1 and pressure goes up, and in addition the gas actually become hotter (temperature rises). But all that is still contained within the volume of gas.

this is most often used to model cylinders in internal combustion engine or air compressors.

for rigid bodies, it is often considered for the whole body itself as kinetic or potential energy gained. As you normally don't tear apart the atoms and that they stay in place, no work is done between them.

Last edited:
etotheipi
Also, just as a further thought, is it possible to show mathematically that internal, non-conservative forces cannot change the total energy of a system?

For internal conservative forces we might say that the work done equals the negative of a change in some potential energy, thus conserving the total mechanical energy.

However, internal non-conservative forces can change the mechanical energy but not the the total energy. I'm not sure how to show this. Perhaps because any such forces will occur in pairs, and their points of application will move through the same displacements?

i think the one of the roots for conservation of energy starts in the
First law of thermodynamics
"For a thermodynamic process without transfer of matter, the first law is often formulated
$$\Delta U = Q - W$$ where ΔU denotes the change in the internal energy of a closed system, Q denotes the quantity of energy supplied to the system as heat, and W denotes the amount of thermodynamic work done by the system on its surroundings. An equivalent statement is that perpetual motion machines of the first kind are impossible."

i think at one point it goes down to the atoms
https://en.wikipedia.org/wiki/Maxwell–Boltzmann_distribution

Last edited:
As far as I know there is no phlogiston, although it does supply a useful discription in a certain realm.
I make the same statement about heat. One can argue that the realm of its utility is larger, but elevating it to something "fundamental" seems the root of much evil. It provides the rug under which we sweep all of the processes we do not wish to consider in detail. And it did work pretty well for Carnot.
But to worry about the fundamental nature of "what is heat" and "what is work" "what is internal energy" etc seems folly to me. It depends upon how you choose to cut the world into subsystems The (mis?)statements of the work energy theorem and the seeming endless arguments so generated derive from arbitrarily truncated definitions of the system and witting or unwitting ignorance of relevant degrees of freedom. They can be "internal" or "external" When those degrees of freedom are not coupled to the problem or are otherwise isolated this sometimes works fine.
For frictional "forces" we choose to ignore the details of atomic motion and interaction at interfaces and then make up a phenomenological force. Small wonder it isn't simple. Amazing how useful it is.
Deformable bodies we can occasionally subdivide into chunks and get away with it (like a spring, for instance). The system complexity goes up as the chunks get smaller.
While this subterfuge provides a useful framework for approximation we should not be surprised when it produces seeming inconsistencies because of the parts left out.

So my answer to the original OP

etotheipi said:
To my mind, there are two distinct approaches to energy problems that different authors tend to use, and I wondered whether either is more fundamental than the other.
They apply to approximations to reality, and each has a locus of utility, neither being "more" fundamental.
But energy is conserved. That is fundamental.

PeroK, jbriggs444 and etotheipi
etotheipi said:
Also, just as a further thought, is it possible to show mathematically that internal, non-conservative forces cannot change the total energy of a system?
No.

The work energy theorem (as applied to point particles) is a mathematical consequence of Newton's second law. In that context, "energy" is defined as ##E=\frac{1}{2}mv^2## and work as ##W=F \cdot ds##.

Now you want to ask about energy conservation for some arbitrary non-conservative force. Before you can do that, you have to come up with a relevant definition for "energy". Does this non-conservative force draw down a supply of electricity from a D cell? Does it burn gasoline from a tank? Does it draw from a pool of magic? If energy is not conserved, it doesn't have to draw from anything at all. Mathematics does not know which until you tell it with an axiom.

cianfa72, hutchphd and etotheipi
jbriggs444 said:
Now you want to ask about energy conservation for some arbitrary non-conservative force. Before you can do that, you have to come up with a relevant definition for "energy". Does this non-conservative force draw down a supply of electricity from a D cell? Does it burn gasoline from a tank? Does it draw from a pool of magic? Mathematics does not know until you tell it with an axiom.

Okey dokey, `what if I define the total energy to be the sum of the mechanical energy and internal energy of the system. More concretely, the sum of the kinetic energies of all the particles in the system and the potential energies associated with each pair of particles in the system.

It is a statement of the conservation of energy that in the absence of external work and heat, this total energy remains constant. For that to be true, the total work done by non-conservative internal forces must be zero.

etotheipi said:
For that to be true, the total work done by non-conservative internal forces must be zero
Because those non-conservative forces are somehow connected to degrees of freedom "outside" the system (might actually be internal and tiny ). This tautological argument is simply one of semantics and not physics. Energy is conserved.

etotheipi
etotheipi said:
is it possible to show mathematically that internal, non-conservative forces cannot change the total energy of a system?

Just in case anyone was still interested, I came up with the following. For any system,

##W_{ext} + W_{int,nc} + W_{int, c} = \Delta T##

##W_{ext} + W_{int,nc} = \Delta T + \Delta U = \Delta E##

Where ##U + T## is the total energy in the system, kinetic plus potential, no matter whether it is microscopic or macroscopic. However, we know from the first law of thermodynamics that in the absence of heat,

##W_{ext} = \Delta E##

So we are forced to conclude that ##W_{int, nc} = 0##, which is the desired result.

This might well be a circular argument, however if the other assumptions are true then it sure seems that internal non-conservative forces can do no net work on a system.

Which is nice, anyway, since otherwise we could have an isolated system whose total energy changes due to internal actions, which is obviously wrong!

hutchphd
etotheipi said:
This might well be a circular argument
I am pretty sure it is. You have assumed conservation of energy in order to deduce that non-conservative internal forces must have an associated reservoir of something that gets lumped in as "internal energy". As a result, these internal forces conserve energy even though they are non-conservative.

etotheipi
jbriggs444 said:
As a result, these internal forces conserve energy even though they are non-conservative.

It was my understanding that for an isolated system (no external work done/heat), mechanical energy is conserved if there are no internal non-conservative forces.

Whilst, also for an isolated system, the total energy (mechanical + thermal) is conserved even in the presence of internal non-conservative forces. Often these will be dissipative, so will funnel mechanical energy into internal energy.

You're right, it's definitely a circular argument. However it does show that if we take conservation of energy in its current form to be true, which I believe is generally accepted, then the total work of internal non-conservative forces must be zero. Even if the contribution of each individual force is non-zero.

etotheipi said:
However it does show that if we take conservation of energy in its current form to be true, which I believe is generally accepted, then the total work of internal non-conservative forces must be zero. Even if the contribution of each individual force is non-zero.
It shows that no matter where Dennis the Menace puts his blocks, there is a number somewhere that accounts for them.

https://www.feynmanlectures.caltech.edu/I_04.html

hutchphd
jbriggs444 said:
It shows that no matter where Dennis the Menace puts his blocks, there is a number somewhere that accounts for them.

https://www.feynmanlectures.caltech.edu/I_04.html

That too

I like to think of the example of a block sliding down a rough ramp, and we might consider the Earth-ramp-block as one isolated system.

The total work done on the system by internal conservative forces manifests itself as (negative changes in) gravitational potential energy, and some other forms (i.e. the potential energy component of some thermal energy, etc.)

But non-conservative internal forces, like the friction between the block and the ramp, will each do some amount of work. The total work done by all forces of this type, however, must be zero.

Like you say, Dennis can't start creating blocks out of thin air.

You can derive conservation of mechanical energy from the work-energy theorem, but you cannot derive conservation of energy. How could it even be possible to do so given the fact that the work-energy theorem follows from Newton's Laws, but Conservation of Energy is a grand principle that includes not just Newtonian mechanics but also thermodynamics? It wasn't until the 19th century that the Conservation of Energy was established, and even as recently as the 1890's it appeared to be in serious doubt because Madame Curie was using rocks (of radium ore) to raise the temperature of water in a calorimeter.

cianfa72, jbriggs444 and etotheipi
By the modern definition of "energy" the conservation of energy follows from time-translation invariance of Newtonian, Minnkowski and static general-relavistic spacetimes. Noether's theorem tells you what energy precisely is within the model. If the symmetry is not realized by the model, you cannot define energy nor is there a conservation law.

Dale and etotheipi
vanhees71 said:
Noether's theorem tells you what energy precisely is within the model.

I came across this whilst reading around, but I think it's going to be a good few years at least until I can understand what it's all about. Something about symmetry under translation in time (?).

Gotta learn the calculus of variations first...

vanhees71
etotheipi said:
It's more because I'm working from Kleppner and Kolenkow, and they gave a proof of the conservation of mechanical energy via the work energy theorem. And quite a few introductory mechanics textbooks sidestep into conservation of energy in the same way, after first integrating Newton's second law.

This seemed a little odd, not least since you mentioned that there are other pathways for transfer of energy within a system.

From a brief look through the relevant chapter I would say that K&K derive the work-energy theorem for a point particle only. And do not generalise it to a system of particles - not that I can see.

The end of the chapter includes a discussion of conservation of energy more generally.

I also found this, which refers to a generalisation to a system of particles (Morin) and includes the additional terms for internal energy:

https://physics.stackexchange.com/questions/247303/work-energy-theorem-for-a-system

etotheipi
PeroK said:
From a brief look through the relevant chapter I would say that K&K derive the work-energy theorem for a point particle only. And do not generalise it to a system of particles - not that I can see.

The end of the chapter includes a discussion of conservation of energy more generally.

I also found this, which refers to a generalisation to a system of particles (Morin) and includes the additional terms for internal energy:

https://physics.stackexchange.com/questions/247303/work-energy-theorem-for-a-system

I also had a look through Morin (and that PSE link) and generally like how he sets the topic out. The only wrinkle is this annoying internal non-conservative work term which I don't seem to know how to get rid of. Even if I take the system to be completely isolated, I can write the change in total potential and kinetic energy as so (taking into account all interactions, microscopic and macroscopic, so no internal-mechanical distinctions yet),

##W_{int, c} + W_{int, nc} = \Delta T \implies \Delta U + \Delta T = W_{int,nc}##

However, we know that for such an isolated system, ##\Delta U + \Delta T = \Delta E## should be zero no matter what, since it is ridiculous to suggest that the system can be gaining energy from internal actions. So the only conclusion is that ##W_{int, nc} = 0##.

Then, I though we could impose an arbitrary classification into macroscopic and microscopic terms in order to setup the "internal energy" term,

##W_{int, nc, mac} + W_{int, nc, mic} + W_{int, c, mac} + W_{int, c, mic} = \Delta T_{mic} + \Delta T_{mac}##

##W_{int, nc, mac} + W_{int, nc, mic} = \Delta T + \Delta U + \Delta E_{th} = \Delta E_{mech} + \Delta E_{th}##

However this doesn't seem to help me that much, since really internal non-conservative forces should be responsible for transfers between ##E_{mech} \leftrightarrow E_{th}##.

This would suggest a relationship like ##\Delta E_{th} = -\Delta E_{mech}##, which is induced by internal non-conservative forces. However, in such a way that ##W_{int, nc} = 0##. This is completely confusing.

I've been playing around for a day or so but wonder if I'm wasting my time. Every undergrad textbook I've consulted on the matter (e.g. Halliday/Resnick) state that mechanical energy of an isolated system is conserved in the absence of internal non-conservative forces, whilst total energy of an isolated system is conserved regardless. I wonder if this just needs to be taken as an empirical fact, since I'm at a loss on how to derive it!

One possible answer is that there are no non-conservative forces.

PeroK said:
One possible answer is that there are no non-conservative forces.

Though I might consider a rigid block sliding across a rough surface, and take the system to be the block and the surface. A frictional force acts on both the block and the surface.

In this example, it is fairly clear that the KE of the block is reducing, so the mechanical energy of that system is reducing. The frictional force of the block on the surface also results in the thermal energy of the surface increasing.

Since the displacements of the two forces are equal, it is fairly evident that the total work done by the two frictional forces (which are internal, non-conservative) is zero. And that ##\Delta E_{mech} = -\Delta E_{th}##.

But we'd need to generalise this somehow.

Or do you mean that all of those non-conservative forces are actually conservative?

To phrase it slightly differently. If there is an internal non-conservative force, then there must be a "fundamental" non-conservative force. I.e. some fundamental interaction that leaks energy out of the universe in some way. That essentially is a necessary condition for (loss of) conservation of energy.

That's why, I believe, Feynman says:

"There is a fact, or if you wish, a law, governing all natural phenomena that are known to date. There is no known exception to this law—it is exact so far as we know. The law is called the conservation of energy. "

Your non-conservative internal force would be an exception to this.

Last edited:
hutchphd and etotheipi
PeroK said:
To phrase it slightly differently. If there is an internal non-conservative force, then there must be a "fundamental" non-conservative force. I.e. some fundamental interaction that leaks energy out of the universe in some way. That essentially is a necessary condition for conservation of energy.

Your non-conservative internal force would be an exception to this.

That's really interesting, I just wonder how then we might rationalise the frictional forces between the block and the surface. If we do break it down into electric interactions between the atoms at the interface between the surface, then we could technically attribute all of that work to microscopic internal potential energy.

I think you're right that a different approach is required. Clearly just fiddling around with algebraic work terms like I've done doesn't give any insight.

If only Dennis the Menace could tell us where all of his blocks are. I think I'll take a break for a bit and try to come back to it with a fresh outlook. I'll try and think on your suggestion.

etotheipi said:
That's really interesting, I just wonder how then we might rationalise the frictional forces between the block and the surface. If we do break it down into electric interactions between the atoms at the interface between the surface, then we could technically attribute all of that work to microscopic internal potential energy.
No because the internal (mobile atoms at surface) degrees of freedom also allow kinetic energy (which we call roughly "temperature") in addition. So both potential and kinetic in "external" Degrees of Freedom. No longer a closed system.
The fact that we can average over the surface DF and get ~simple reproducible results lulls us into thinking it is "just another force". It is not intrinsically so, but the concept has some utility. Pedagogically it is a disaster IMHO..

etotheipi
etotheipi said:
Though I might consider a rigid block sliding across a rough surface, and take the system to be the block and the surface. A frictional force acts on both the block and the surface.

In this example, it is fairly clear that the KE of the block is reducing, so the mechanical energy of that system is reducing. The frictional force of the block on the surface also results in the thermal energy of the surface increasing.

Since the displacements of the two forces are equal, it is fairly evident that the total work done by the two frictional forces (which are internal, non-conservative) is zero. And that ##\Delta E_{mech} = -\Delta E_{th}##.
As far as I can understand, your claim is the following:

Take the system 'block+surface' as closed (no external work/heat exchanged); mechanical energy of this system is just KE. Frictional forces (internal, non-conservative) act in a couple: the former acts on the block and latter on the surface.

If the total work done by the two frictional forces was zero there shouldn't be any change of KE (assumed to be the total mechanical energy of the 'block+surface' system).

Thus, IMHO, we shouldn't assume the 'block+surface' as a 'closed' system. Said that in other words: considering 'non-conservative' forces acting on parts of a system implicitly assume a 'non-closed' system

etotheipi
cianfa72 said:
Thus, IMHO, we shouldn't assume the 'block+surface' as a 'closed' system. Said that in other words: considering 'non-conservative' forces acting on parts of a system implicitly assume a 'non-closed' system

I suppose that's one way of viewing it also.

I'd still be inclined to say that the system is still closed, so long as we account for the thermal energy evolved. Such that ##T_{block} + E_{th} = \text{constant}##. Since the frictional forces do seem to be internal to that system.

Though I know @hutchphd mentioned in another thread that we can treat energy dissipated due to friction as outside the configuration space. So the interpretation you propose appears to be just as correct.

etotheipi said:
I'd still be inclined to say that the system is still closed, so long as we account for the thermal energy evolved. Such that ##T_{block} + E_{th} = \text{constant}##. Since the frictional forces do seem to be internal to that system.
ok, thus coming back the to your OP question: saying ##T_{block} + E_{th} = \text{constant}## means to say that 'general' principle of conservation of energy (COE) cannot be 'derived' from work-energy theorem/principle (WEP)...basically including non-conservative forces we are not able to introduce an energy (potential energy) as a 'rescue' for the work-energy theorem to be always true (better to say applicable)

etotheipi
If the system is closed then energy of the system is conserved.
If the system is not closed, then to track the energy one must characterize the leaks.

This seems sufficient, and I do not understand what more needs to be said.

vanhees71 and etotheipi
I have not read the thread carefully, and surely what I am going to say is clear for everybody but I nevertheless want to recall this.
Classical mechanics systems are described by the following equations
$$\frac{d}{dt}\frac{\partial L}{\partial \dot x^i}-\frac{\partial L}{\partial x^i}=Q_i,\quad L=L(t,x,\dot x)\quad i=1,\ldots,m.\qquad (*)$$
The work energy theorem is written as follows
$$\dot H=-\frac{\partial L}{\partial t}+Q_k\dot x^k,\qquad (**)$$ here
$$H=\frac{\partial L}{\partial \dot x^i}\dot x^i-L$$ is the energy of the system.
One may also rewrite formula (**) in the integral form.

And formula (**) is deduced from the equation (*). And it is clear what one should demand for the energy to be conserved: ##\dot H=0.##

vanhees71, hutchphd and etotheipi

• Mechanics
Replies
2
Views
986
• Mechanics
Replies
19
Views
1K
• Mechanics
Replies
6
Views
2K
• Mechanics
Replies
9
Views
2K
• Mechanics
Replies
4
Views
852
• Introductory Physics Homework Help
Replies
8
Views
372
• Mechanics
Replies
10
Views
1K
• Mechanics
Replies
1
Views
1K
• Mechanics
Replies
8
Views
3K
• Mechanics
Replies
4
Views
1K