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Aside from Noether's Theorem, how do we know energy is conserved?

Energy is the capacity of a system to do work. It's the number that tells me how much "force" a system can apply over a distance. For long I knew we could create forces. However, the only reason... The physical reason that conservation of energy is making sense to me is that when I apply a force to move matter from point $A$ to $B$, I must apply force $x$ on it over displacement $D$ in order to return it I must apply the same amount of force $x$ over $D$. Also the fact that all our natural forces are conservative (in most cases), if work is done by any of the four forces. You must apply the same work again because the natural force is acting against you. Example: Ball is attracted down to Earth's surface by gravity, $9.8$J of work is done ($mgh = 1 \times 9.8 \times 1$) $9.8$N of force has been applied. When the ball is at rest on the surface, I must apply the same amount of force against gravity to move it upwards back to point $A$ ($1$m above the surface). Does this make any sense?

Why is energy conserved besides Noether's Theorem? What other physical observations show it must be true in all our current observable systems?

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You should start with a more basic question. Something like "why does the universe exist?" Joking aside, how do we know that any of our mathematical models are "true"? The only thing we can do is to verify them with experimental observation to the best of our limited ability. –  SimpleLikeAnEgg Oct 31 '13 at 17:42
    
Our observations are based of our numbers aren't they? What proved to "us" that energy is conserved? What showed that "number" to be the same always in a system? –  M. A. A. Nov 1 '13 at 3:08
    
Physics is not about proofs in the way of Quod Est Demonstrandum . Observations and experiments lead to trends and correlations. Then a brilliant physicist ( like Newton) devises a mathematical model, the mathematics of which display consistency and QED and the model is validated by the experimental observations. Then physics has advanced a step. When experiments delve further the need for new mathematical models, overlapping or merging with the previous ones, are developed and validated ( example quantum mechanics) –  anna v Nov 1 '13 at 4:24
    
I wonder, we observed systems that demonstrate conservation of energy, why wouldn't we interest ourselves in systems that violate that theory? –  M. A. A. Nov 1 '13 at 7:21

2 Answers 2

You start with a Lagrangian for the system. If it is invariant under temporal translation (keeping everything else the same), using Noether's theorem, you get a function that is constant with respect to time (energy).

But you can't do squat with Noether's theorem without first obtaining a Lagrangian from physical observations (Newton's Laws).

So Noether's theorem requires physical observation to work its magic, like all good mathematics used in physics. Don't let anyone tell you otherwise.

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Yeah, that does make sense. Newtonian law's makes perfect sense to aiding conservation of energy. –  M. A. A. Nov 1 '13 at 2:51

Conservation of energy started as an experimental observation. Wikipedia has the history of how scientists of the time arrived to the concept of conservation of energy as a concept.

It was Gottfried Wilhelm Leibniz during 1676–1689 who first attempted a mathematical formulation of the kind of energy which is connected with motion (kinetic energy). Leibniz noticed that in many mechanical systems (of several masses, m_i each with velocity v_i ),

enter image description here

was conserved so long as the masses did not interact. He called this quantity the vis viva or living force of the system. The principle represents an accurate statement of the approximate conservation of kinetic energy in situations where there is no friction.


In a paper Über die Natur der Wärme, published in the Zeitschrift für Physik in 1837, Karl Friedrich Mohr gave one of the earliest general statements of the doctrine of the conservation of energy in the words: "besides the 54 known chemical elements there is in the physical world one agent only, and this is called Kraft [energy or work]. It may appear, according to circumstances, as motion, chemical affinity, cohesion, electricity, light and magnetism; and from any one of these forms it can be transformed into any of the others."

It was a long drawn out observational process where many contributed.

The mathematical formulations came and made empirical observation into a strict law, as explained in the other answer.

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Yes, but what are those observations? If you spend something to do something, you need to spend the same amount again? Which makes sense to my observations. Energy is conserved because of the -forces acting on against what ever is being done. If there are no equal negative forces energy is created in some sense. –  M. A. A. Nov 1 '13 at 3:00
    
If you read the article you will see that the early thinkers followed similar routes. The conservation laws as absolutes were crystallized in the form we have after the mathematically strict models were developed , not before. –  anna v Nov 1 '13 at 4:26
    
It stunning really... It's really about forces isn't it? I mean, conservation of energy could not be true if there is no +force = to -force. The work we do on something is really a +force to move it or change it + -force acting against us. It's the word "conservation" that just didn't make sense to me until I started to study conservative fields. And seeing how energy and forces are linked intimately. –  M. A. A. Nov 1 '13 at 7:20

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