Is conservation of energy derived from the work energy theorem?

[etotheipi]

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.

 

[Herman Trivilino]

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.

 

[vanhees71]

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.

 

[etotheipi]

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...

 

[PeroK]

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]

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,

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

This leads to

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

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!

 

[PeroK]

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

 

[etotheipi]

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?

 

[PeroK]

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.

 

[etotheipi]

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.

 

[hutchphd]

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..

 

[cianfa72]

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]

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.

 

[cianfa72]

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)

 

[hutchphd]

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.

 

[wrobel]

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: $\dot H=0.$