# Conservation of energy, (lack of) proof

## Main Question or Discussion Point

Thermodynamics is clearly based on the assumption that energy is conserved. I ended up discussing this assumption with a fellow student, who states that this assumption can be rigidly justified and means that Noether's theorem proves this.

Everywhere I have read about it, the conservation of energy have been expressly confirmed in experiments and observations, but there is no way to prove this conclusion.

I would like to hear your thought on this subject (aside from that digging deep enough, all scientific facts are based on theories that have been sucessful in predicting experimental results, in some sense.)

My point is that conservation of energy is an assumption that cannot be rigorously proved, while he implies it is completely justified, and keeps referring to the above theorem, no matter what arguments are presented.

Thanks :)

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russ_watters
Mentor
Everywhere I have read about it, the conservation of energy have been expressly confirmed in experiments and observations, but there is no way to prove this conclusion.
That sentence is self-contradictory: that's what proof is in science.

Well, I am aware of that, even though I formulated that sentence very badly, not being sure what other word to use to eliminate the ambiguity. (I am sorry about that). In fact, that is why I wrote "aside from that digging deep enough, all scientific facts are based on theories that have been sucessful in predicting experimental results, in some sense". In other words, "a proof" didn't mean a scientific proof, but rather an "everyday proof", in which case saying "Every experiment so far has confirmed it, but we can't with 100 % certainty say that it is applicable to every system in the universe..." is not a proof.

Of course nothing about science is certain. What sucessful theories have in common is that they are confirmed by experiments.

The post is rather about what uncertainties there are in the assumption of conserved energy. What systems could we possibly observe to find contradictory results? And to limit ourselves, could there be such systems that are not strictly theoretical? Which of course is a very poorly worded question.

russ_watters
Mentor
Since conservation of energy is users everywhere, always, there are countless challenges to it and opportunities for it to fail happening all the time.

I'm really not sure seeing your point/question/concern.

DrClaude
Mentor
Have a look at the history of the neutrino. When it turns out that it is better to propose a new particle noone had observed before instead of ditching the conservation laws, that is a pretty strong ground for the conservation laws.

And as was pointed out to you, Noether's theorem shows that conservation of energy is just a consequence of the invariance of physical equations with respect to time translation.

Dale
Mentor
From the basic laws of physics as we know them, conservation of energy can be derived. So conservation of energy is not assumed, it is derived. What is assumed are the laws of physics.

A violation of the conservation of energy would mean that one or more of the laws of physics is wrong, which would be testable in many different ways, depending on which law were wrong. If you have a specific law in mind then we can discuss what would happen experimentally if that law were wrong in a way which led to violations in the conservation of energy.

Jano L.
Gold Member
The law of conservation of energy in general sense is an idea derived from experience by Robert Mayer, Joule and Kelvin and others; they found that there is certain equivalence of work and heat which allows us to introduce the notion of internal energy, which is constant for isolated system.

The idea is accepted in its generality without proof - it is a basic law (1st law of thermodynamics). In particular cases, a mathematical proof is possible based on a model of the physical system, for example system of differential equations of motion. The Newtonian equations of motion and the Maxwell equations have certain integrals of motion which lead to definition of energy and its continuous conservation.

The proof is based on the validity of the those equations. However, there are other kinds of equations of motion (including friction, delays, etc.) that do not have such integrals of motions. So far the prevalent opinion is that the above kind of equations is basic one and the other kind is only effective description of certain complicated phenomena which are hard to analyse based on the Newtonian and Maxwellian equations alone.

Experimentally, there is no single evidence of violation of conservation of energy. But still, there may be such violation, we just have not discovered it yet.

Erland
This is a very interesting issue.

My opinion is that the conservation laws for energy, linear momentum, angular momentum, etc. are not empirical statements, but assumptions, more or less axiomatic in nature. Sometimes, phenomena are discovered which seem to contradict some of these conservation laws, but then, the physicists introduce some new term in the relevant conservation equation, and call it "X-energy" or something like that, and then the conservation equation holds agian.

A typical example is that the classical law of conservation of linear momentum does not hold for electromagnetic phenomena (for example, if two charged particles are moving with constant speeds in directions perpendicular to each other with one of them passing directly in front of the other one, then a magnetic force is exerted by the first particle upon the second, but not the other way round, thus changing the mechanical linear momentum). The physicists solved this by saying that the difference in linear momentum which arises constitutes an "electromagnetic linear momentum", which, if it is plugged into the conservation equation, makes it hold again. This electromagnetic linear momentum is supposed to be carried by the electromagnetic field.
Likewise, one has introduced electromagnetic energy and electromagnetic angular momentum. The same method is regularly applied when a phenomenon is dicovered which seems to contradict some conservation law.

This method is possible to apply as long as the difference between the sides in the relevant equation is some way regular, possible to express in terms of the quantities in the equation. Then, this expression for the difference is defined as a new type of energy (or linear or angular momentum, etc.). This must be considered as a sound method as long as we have this kind of regularity, and the expressions not become too complex.

If, in the future, this method needs to be applied over and over again and the new types of energy etc. become increasingly complex and unintuitive, then it is perhaps time to abandon the conservation law and search for an alternative way to formulate and understand the relevant parts of physics. But hitherto, nothing indicates that this will happen.

Nugatory
Mentor
My opinion is that the conservation laws for energy, linear momentum, angular momentum, etc. are not empirical statements, but assumptions, more or less axiomatic in nature.
You've left out a third possibility: They are neither empirical statements nor assumptions, but theorems proved from more basic assumptions about the universe - google for "Noether's Theorem".

Historically, of course, the conservation laws have been all three, at different times. They started as empirical statements ("Look at all these observations suggesting that some quantity is conserved"); stood up to the test of time so well that people came to assume their truth (and started calling them "laws"); and now we consider them theorems derived from more basic assumptions about the symmetries of the universe.

• 1 person
I really appreciate all your answers, but I was hoping for it to bring up some kind of discussion of your thoughts and not restatements of the laws of physics and how they are based on fundamental observations and laws of nature. Once again, I would like to state that I wrote "aside from that digging deep enough, all scientific facts are based on theories that have been sucessful in predicting experimental results, in some sense", which I would say implies clear awareness of that there exist fundamental laws on which the laws of physics are based and that some laws are clearly (mathematically deduced) consequences of these.

Erlan's answer comes the closest to what I am looking for here, expressing an interest in the discussion and presenting some aspects related to the conservation of energy, and the nature of physical laws in general.

I am sorry for rejecting most of your answers, but I feel that there is a need to guide the discussion in the "right" direction, which means that we can't just accept the parameters of Nother's theorem.

I am NOT arguing that energy is not conserved, 'cause I am, like most people are, convinced that energy is conserved. Instead of studying exactly all aspects of the most fundamental physics, (which I suppose is impossible in a reasonable amount of time). I am hoping there might pop up some good thoughts from some imaginative people around the world.

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Drakkith
Staff Emeritus
Conservation of energy has been observed to apply every single time we've looked. Furthermore, for energy to not be conserved, something VERY strange would have to occur, such as particle NOT reacting when a force is applied, or moving without a force being applied.

I'm trying to decipher your posts, but it seems you've typed a lot without really saying anything at all. What exactly are you hoping to have discussed here? "Aspects related to the conservation of energy, and the nature of physical laws in general" is incredibly broad and non-specific, something which doesn't encourage much of a discussion since readers have no guidance on a specific topic. (Especially after you've squashed the discussion that DID happen)

Dale
Mentor
I am sorry for rejecting most of your answers, but I feel that there is a need to guide the discussion in the "right" direction, which means that we can't just accept the parameters of Nother's theorem.
You cannot reject Noethers theorem. It is a mathematical proof. There is no room for accepting or rejecting it, it is mathematically proven. If you do not accept Noethers theorem then you have to reject all math. That is not a logically tenable position and certainly not an acceptable direction for discussions on this forum.

The only thing that you can do is find instances where Noethers theorem doesn't apply.

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Erland
Conservation of energy has been observed to apply every single time we've looked. Furthermore, for energy to not be conserved, something VERY strange would have to occur, such as particle NOT reacting when a force is applied, or moving without a force being applied.
But that is impossible by definition (at least if we change "moving" to "accelerating"), for "force" is defined as something that gets objects to accelerate.

If we would find a particle which accelerate without any known force is being applied, we would, by the present paradigm, assume that some unknown force is applied, and that there exists some up to that time unknown type of energy, just as was done in the electromagnetic case I mentioned in my previous post, except that it was about linear momentum, not energy.
And this method has worked find for along time, so we have no reason to change this.

Erland
You've left out a third possibility: They are neither empirical statements nor assumptions, but theorems proved from more basic assumptions about the universe - google for "Noether's Theorem".
I am not so impressed by these invariance principles. For example, the law of conservation of mechanical linear momentum can be derived from an invariance principle w.r.t. spatial translations. To carry through this derivation for a system of particles, we need to assume that the forces between the particles can be derived from potentials which only depend upon the relative positions of the particles (and not e.g. time and the velocities).
On the other hand, just by assuming Newton's third law (in the weak form), we can prove conservation of mechanical linear momentum without assuming any potentials at all.
So at least for mechanics, conservation of linear momonetum seems to be a more general principle than this invariance w.r.t. spatial translations. Can't really say how this applies in other fields of physics, though.

Also, moving from the conservation laws to the invariance princples only moves the problem to the lagriangian and the potentials instead of energy (in general), and linear and angular momentum. If we find a new phenomenon where some conservation law seems to be violated, we still have to find a new type of lagrangian for the problem.

WannabeNewton
I am not so impressed by these invariance principles...Can't really say how this applies in other fields of physics, though.
It's very unfair to make that first claim in light of the last isn't it? Noether's theorem is much more fundamental than classical mechanics.

Erland
It's very unfair to make that first claim in light of the last isn't it? Noether's theorem is much more fundamental than classical mechanics.
How can it be unfair to say how I feel? This wasn't a claim about objective reality.
And it is possible that I change my mind when I learn more.
And, as far as I can see, there are also parts of classical mechanics that are not covered by Noether's Theorem.

UltrafastPED