For example, do we find something related to the modern energy concept in Ancient China, Ancient India, or the Islamic Golden Age?
Among "similarities and differences", conservation is obviously important, as is convertibility and variety of forms.
If the word energy or a close cognate was not used, what is the justification for regarding it as a related concept?
Etymology
Quite a few sources say that Aristotle was the progenitor of the term, after his use of energeia (a Latin transliteration of ἐνέργεια). However, one source notes that Heraclitus used a similar word, en-ergon, years prior. Heraclitus wrote:
"En-ergon is the father of everything, king of all things and, out of it, all forms of contrast originate. Since ‘en-ergon’ is common to everything, it is vital for life itself.”
This was part of his aforementioned flux theory. However, like Aristotle's ideas, his concepts bear no relation to our modern usage of 'energy', and I doubt the two are related. It is worth noting, though, that Thomas Young's coinage of 'energy' in the early 19th century is thought to have been influenced by Aristotle's usage of energeia. This would mean that the two, while not the same idea, may be thought of as true cognates.
Comparison
I’m going to compare some ancient ideas and explain why they aren’t really precursors to our modern idea of ‘energy’. I suppose this is a "Too long; didn't read" summary; a lot of the other things are extraneous. Note: All information is taken from sources given later in the answer., unless otherwise indicated.
After a week or so of research, I found just about nothing that directly confirms or denies the possibility presented in the question. Zip. Zero. Nada. I've gotten the only null result I've ever gotten in 4 months when researching an answer. It's frustrating, but it's a bit helpful, because it taught me that there are some things that you may never know, no matter how hard you try to find them.
Back to the null result. The question asked whether or not any non-European, ancient cultures came up with something related to our modern concept of energy. The answer, as far as I can tell, is that nothing of the sort existed before a few centuries ago.
If you'll pardon this little detour, I'll talk about Europe first. As fdb hit upon, Aristotle is often cited as the creator of a concept similar to energy. I discussed it in this answer. The problem is (which I should have expanded upon), actuality and potentiality were philosophical ideas. Sure, loose interpretations could make them seem similar to the "energy" of today, but they were quite different from our modern concept. By Aristotle's logic, a tree resting at what we would call a reference point (i.e. where potential energy is 0) would still have potential - the potential to become a wooden bowl!
Next, we go to Gottfried Leibniz and the idea of vis viva (behind a paywall and/or registration, but the free first page should give you an idea of what I'm talking about). He argued that a moving object was imbued with some sort of quantity. This could be interpreted as either kinetic energy or momentum, and in fact there was a dispute over this. We might also say that Descartes was also involved. Nevertheless, 350-ish years ago, there was some idea (in Europe) that a quantity in moving object was conserved.
Going to Wikipedia (a short and mostly irrelevant detour, I assure you), we find that the term "energy" (and then "potential energy" and "kinetic energy") was not coined until the early 19th century, by a few separate scientists working on the idea. At this point, we may consider the concept of energy (in its modern incarnation) to have been born. If you're fine with Wikipedia, you can read more here.
Try a few searches (duckduckgo is my engine of choice) on the internet. I recommend the following:
. . . and many more. All turn up results either about energy generation or (once or twice) crank sites about weird life-force ideas that are decidedly kooky. I was not able, in fact, to get a single mention of ancient cultures when it came to the concept of energy. This is pretty bad, because a result of "No, there was no idea of energy in ancient cultures" would be definitive. Here, there is nothing.
The one exception I found was from this site, which gave the completely following (no sources were cited):
Ancient cultures associated fire with all forms of energy, and the Greek philosopher Heraclitus (~500 BCE) explained everything in the universe in terms of energy. In the centuries that followed, many scientists proposed theories to explain the concept of energy, but few shed any true light on the subject.
Unfortunately, the "ancient cultures" are nowhere to be found. Go ahead a couple hyperlinks to this page. Heraclitus' flux theory is mildly interesting but not too relevant. Further down, we see hints of something along the lines of conservation of mass, but it goes nowhere. The knowledge that Aristotle disagreed with him leads me to believe that there is no hope of a connection between the two.
Just for fun, I went to the Wikipedia page on Qi. The on thing that caught my eye was this sentence:
Notions in the West of energeia, élan vital, or "vitalism" are purported to be similar.
Going to the pages listed reveal nothing of any importance (and none of the ideas are anything like our modern idea of energy), but the fact that Aristotle is listed nearby further convinces me that his work is not really scientific.
One last stop. Many searches brought me here, where the author (see the PDF link on the upper left; I've had a hard time copying it here) claims that Joule and Mayer were the originators of the idea of energy. It seems like they certainly made some empirical investigations, but the ideas of Descartes and company seem to predate these men.
I'll end with Bertrand Russell. For those who don't know, Russell's Teapot is the idea that the burden of proof is on the claimant. I could say that there is a teapot floating between the Earth and Mars; it's nearly impossible to disprove unless you search every corner of the space between the solar system and so I can say that it exists. Our situation is similar to that, in that the idea that ancient cultures had a concept similar to energy is hard to disprove. One the one hand, I have found nothing; on the other, there are still chances that I am completely wrong. I'm going to let the community judge this answer and its validity, but at the moment, my answer to the question is "No."
By the way, I'd love some positive and/or negative feedback here. This was an odd experiment for me, and I'm not sure how well it worked.
In ancient Indian texts like Vedas and puranas there are so called philosophies or concepts of energy. Actually vedas and puranas are considered as religious texts, but these texts contain many hidden facts and knowledge(I doubt how many of you agree with this statement. Some of the examples of science in Vedas are, Vedic mathematics, Astrology, Vasthushasthra, Yoga and many more ). It will be broad to compare the philosophies in Vedas and modern science energy theories in this small answer and also we need experts on both to verify them. Even though there are thousands of websites discussing the science in Vedas which also includes the comparison of energy concepts.
Let's check this example,
Atom bombs and thermonuclear bombs : powerful weapons which are practical examples of modern energy concept.
So was there an equivalent weapon concept anywhere in the world? Yes, there was and it was in Ancient India. See this google search result and a quote from wikipedia is below.
Modern day Nuclear weapon and Brahmastra
Robert Oppenheimer (1904-1967) was a scientist, philosopher, bohemian, radical, fanatic of ancient Sanskrit literature,a theoretical physicist and the supervising scientist of the Manhattan Project, and most importantly, the developer of the atomic bomb. Seven years after the first successful atom bomb blast test in New Mexico (Trinity), Dr. Oppenheimer was giving a lecture at Rochester University. To the question “Was the bomb exploded at Alamogordo during the Manhattan project the first one to be detonated?” he gave a strange reply “Well — yes. In modern times, of course.” And as for Oppenheimer’s first words after the detonation of the bomb he quoted from Hindu epic Mahabharata , “If the radiance of a thousand suns were to burst at once into the sky, that would be like the splendor of the mighty one. Now, I am become Death, the destroyer of worlds.” (Bhagavad Gita)
The previous quote is for praising Lord Krishna when he displayed his real figure to Arjuna, and this has nothing to do with the energy concept. But we can conclude that Dr.Oppenheimer was familiar with the Bhagavat Gita and the book had it's influence on him which made him say those words from the book. (It is to be noted that, Bhagavat Gita includes messages in the middle of the Mahabharatha war. During this war people could have used the Brahmasthra, Pashupasthra or similar kind of weapons.)
Most people agree that no human civilization before us had knowledge of atomic energy and its by-products. The atomic bomb is something completely novel to modern science. But we find in the Vedic literature descriptions of weapons that had a similar amount of energy as the atomic bombs we use today. “The atomic energy fissions the ninety-nine elements, covering its path by the bombardments of neutrons without let or hindrance. Desirous of stalking the head, i. e. the chief part of the swift power, hidden in the mass of molecular adjustments of the elements, this atomic energy approaches it in the very act of fissioning it by the above-noted bombardment. Herein, verily the scientists know the similar hidden striking force of the rays of the sun working in the orbit of the moon.” (Atharva-veda 20.41.1-3).2
See the full wikipedia link. From this link it can be understood that there were concepts like these before thousands of years ago in Vedas(which are older than 5000 years ago). Nobody added the Brahmasthra concept just to compare it with modern atom bombs, it was there in the ancient text and Hindu puranas discuss many of such weapons in different stories. Did these weapons used in ancient days? The verification is difficult but one thing is very much sure, the idea was there in these ancient Indian texts much earlier.
Many philosophical theories in Vedas and puranas are comparable with modern theories. The May be we all can wonder from where the so called modern energy concept was derived. The actual origin of the modern science energy concept could be from these ancient knowledge. Many scientists like Albert Einstein , Neils Bohr and Oppenheimer were admirers of these ideas and there would have been impact of these ideas in their thought process of deriving modern energy concepts(If there is a verification for the same, we can not actually call it 'modern energy concept', we should call it as just the 'Energy concepts').
Another wiki link about Heisenberg who proposed uncertainty principle, see the following quote in the wiki link (not discussing the ideas which influenced him as I am not an expert in both subjects, also I was not present during that discussion:). Just mentioning to prove that such concepts had influenced scientists.)
After these conversations with Tagore some of the ideas that had seemed so crazy suddenly made much more sense. That was a great help for me. On conversations with Rabindranath Tagore, as quoted in Uncommon Wisdom: Conversations With Remarkable People (1988) by Fritjof Capra, who states that after these "He began to see that the recognition of relativity, interconnectedness, and impermanence as fundamental aspects of physical reality, which had been so difficult for himself and his fellow physicists, was the very basis of the Indian spiritual traditions." Variant: After the conversations about Indian philosophy, some of the ideas of Quantum Physics that had seemed so crazy suddenly made much more sense.
Here are some basic links which I found in my search.
https://cpdarshi.wordpress.com/2011/11/22/modern-physics-found-its-direction-from-vedanta/ http://www.krishnapath.org/quantum-physics-came-from-the-vedas-schrodinger-einstein-and-tesla-were-all-vedantists/ http://www.hinduwisdom.info/Advanced_Concepts.htm
Weight is an old concept related to energy that was discussed both in not-so-Ancient China, Ancient India, or the Islamic Golden Age.
It meets each and every of your criteria :
I will document this answer latter as I lack knowledge of Ancient China and Ancient India, but I'm sure I can document most aspects for the three given civilisations.
Our word “energy” is borrowed via Latin from the Greek energeia, a term coined by Aristotle from the preposition en (“in”) and ergon (“work”). He developed this concept in his discussion of potentiality versus actuality. For “potentiality” he used the word dynamis “capability”, while for “actuality” he used two apparently interchangeable terms energeia and entelecheia. (The identity of the two terms is expressed by Aristotle is his Metaphysics 1047a30: ἐλήλυθε δ᾽ ἡ ἐνέργεια τοὔνομα, ἡ πρὸς τὴν ἐντελέχειαν συντιθεμένη.) Is this sense dynamis overlaps largely with the modern concept of “potential energy”, while energeia is perhaps rather what we would call “kinetic energy”. Aristotle expresses the difference between the two rather clearly in this passage from Metaphysics 1017a (Tredennick’s translation):
“For we say that both that which sees potentially and that which sees actually is ‘a seeing thing’. And in the same way we call ‘understanding’ both that which can use the understanding, and that which does ; and we call ‘tranquil’ both that in which tranquillity is already present, and that which is potentially tranquil. Similarly too in the case of substances. For we say that [a statue of] Hermes is in the stone, and the half of the line in the whole; and we call ‘corn’ what is not yet ripe.”
But you asked about pre-modern non-European cultures. To this we can reply that the mediaeval philosophers in the Islamic world were definitely familiar with the Aristotelian concepts of potentiality versus actuality, but for the former they did not (as far as I can see) use the word energeia, but they did use its synonym entelecheia, which was borrowed directly into Arabic and spelt in Arabic as إنطلاخيا. In this sense you can say that mediaeval Islamic philosophy had a concept of “energy”, but did not use the term “energy” as such.
