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I am thinking in the mechanical context.

Everywhere I research (e.g. Wikipedia) the law of conservation of energy is stated only for the special case of an isolated system. I am wondering if conservation of energy holds beyond that special case; it seems that it should. After all, if a property holds only under special cases then the property is not a law.

Reading Feynman's lecture 14 volume I, I understand that if only conservative forces act on an object its total energy remains unchanged. For example, a falling object subject only to gravity has a constant sum of kinetic and potential energies. However, the system consisting of just this object is not an isolated system because it is subject to the external force of gravity. It seems this is an instance of conservation of energy holding outside the special case of an isolated system.

Feynman argues that at the fundamental level all forces are in fact conservative. This implies that at the fundamental level conservation of energy applies to all systems. Is this true? If so, why is conservation of energy continually stated under the special case of an isolated system?

(this site's "energy-conservation" tag says "the amount of energy in a system is constant" implying the system need not be isolated, further confusing me)

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  • $\begingroup$ Technically, energy should not be perfectly conserved in the potential + kinetic energy of a falling object. As the object falls, the earth also accelerates toward it and gains a tiny bit of kinetic energy, which is almost always negligible. This emphasizes, as others have said, that to have perfect energy conservation one must either have an isolated system or keep track of the energy that is flowing out. $\endgroup$
    – Rococo
    May 15, 2016 at 1:14

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There are different ways of stating conservation of energy and accounting for energy, which can make the issue confusing. One such statement is "the total energy of an isolated system is constant". This is true, and is the simplest way to state conservation of energy. This form of conservation of energy is the earliest taught.

There's another way of stating conservation of energy, "the energy in a region changes by the amount of energy flowing into or out of a region, and energy in adjacent regions changes by the same amount". You could call this local conservation of energy, and is a much stronger statement. It not only tells us that energy is conserved, but it also tells us that energy can't disappear from a region and reappear far away. This is the kind of conservation of energy that Feynman is considering, so he can apply it to systems that aren't isolated.

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  • $\begingroup$ Suppose the falling object and the earth are two adjacent regions. Considering the only difference between these regions is mass, which way does the energy flow? Both regions should gain kinetic and lose potential energy. $\endgroup$
    – Joseph Johnston
    May 15, 2016 at 15:15
  • $\begingroup$ For that case it's thought of as energy flowing from the gravitational field to the objects. The fields themselves are where the potential energy is stored. One thing though is this kind of thinking isn't applied as often to gravity, since energy doesn't generalize well to higher study of gravity (general relativity). $\endgroup$
    – David
    May 15, 2016 at 17:38
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Energy is conserved in an isolated system. The energy of the system is constant. But it can flow from one part of the system to another.

Energy is not conserved in the part. The energy in the part can increase or decrease. No energry is created or destroyed. It just moves to a place where you stop counting it.

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  • $\begingroup$ Suppose the falling object and the earth as an object form an isolated system. Which way is the energy flowing between these two parts? One part simply has more mass than the other, and both parts seem to gain kinetic and lose potential energy. $\endgroup$
    – Joseph Johnston
    May 15, 2016 at 15:14
  • $\begingroup$ You are right. In this case, it is just converted from one form to another. But you should not think of energy something in an object. It is more like an accounting system. See this. physics.stackexchange.com/q/94281/37364 $\endgroup$
    – mmesser314
    May 15, 2016 at 21:05
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Feynman knows better than that, actually, and this is one of these cases were undergrad textbooks like Feynman's come dangerously close to being not only incomplete but deceptive. Conservation of energy is a consequence of time translation symmetry (see Noether's theorem) and he mentions it in chapter 52 without even an attribution (bah!). If that holds, then the question of open vs. closed systems is merely an accounting issue, as @mmesser314 said. If it doesn't hold, and there are signs that it may not hold on cosmological scales, then energy is not conserved, period.

I never liked Feynman's books. While he has a lot of really nice examples and detailed explanations (too detailed, sometimes), he misses to give students the big picture first. There is no law against starting with the most important stuff in physics first, which is what Russian textbooks like Landau-Lifshitz do. They expect you, the reader, to be intelligent and to be able to swim once they throw you into the water. The advantage of that is that you are not being told that the waters you are in are shallow and full of little waves like in Feynman's baby pool. To be honest... don't study from Feynman, alone. Get yourself a book for "real physicists" like Landau-Lifshitz. Throw yourself into the pool and you will find that you won't drown.

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  • $\begingroup$ Well, Curious, I am with you on Feynman and his personal view of physics. I was lucky to be able to hear him lecture back in 1964 in a school at Erice, and then in ~1981 in a workshop in Crete. He had a very special pov of physics, which does not coincide with my observer's post/seat. Analogy: If am told "touch your left ear with your right hand" I pass my right hand from front to the ear. It seemed to me that Feynman would pass it behind his head. Mind you I am talking of discussing what was brand new physics at that time. That is why I never read his lectures. This does not diminish his $\endgroup$
    – anna v
    May 15, 2016 at 5:07
  • $\begingroup$ brilliance, which must also depend on his ability of thinking outside the box . It is an observation that physicists have differing mental processes and in order to progress in studies fast, it is good to latch onto books and presentations that resonate with one's own thinking. $\endgroup$
    – anna v
    May 15, 2016 at 5:09
  • $\begingroup$ I completely agree. Feynman was capable of some of the best science writing of the century. I get what he is trying to do in his lectures, but IMHO he fails his audience by setting the level too low and in this case, I would say, he commits a major sin. Emmy Noether deserves better at the introductory college level. $\endgroup$
    – CuriousOne
    May 15, 2016 at 6:24
  • $\begingroup$ @DIYser: Can I give you a rain check tonight? I have to support someone traveling in Asia and the time zone difference is not favorable... :-) $\endgroup$
    – CuriousOne
    May 15, 2016 at 7:22
  • $\begingroup$ @CuriousOne Interesting explanation. So conservation holds if and only if symmetry of time translation holds. Thanks for the textbook recommendation. I agree with that learning approach. $\endgroup$
    – Joseph Johnston
    May 15, 2016 at 15:13

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