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In classical mechanics, is there any way to prove that energy is conserved without Noether's theorem and also without assuming Newton's laws?

When I ask IF something is possible, I'm not claiming it is, I'm just asking because I'm curious. I would say sorry for not having a major in physics and not knowing this obvious thins but I'm sadly not.

Suriya
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  • Yes, there is: the theorem of conservation of energy can be proved directly from Newton's second law. – Massimo Ortolano Oct 14 '16 at 15:15
  • @MassimoOrtolano I don't think you can, at least without assuming third law too. Taking F=ma as your hypothesis you could only derive by integration $KE+PE=0$ and deriving it to see if the derivative equals $0$ you'd get again to your hypothesis, F=ma, so nothing is proved. – Suriya Oct 14 '16 at 15:20
  • Yes, you need also the third law. – Massimo Ortolano Oct 14 '16 at 15:22
  • @MassimoOrtolano That should be expanded into an answer – David Z Oct 14 '16 at 15:22
  • @MassimoOrtolano yes but I said if there is without third law (or conservation of momentum) – Suriya Oct 14 '16 at 15:23
  • For how to prove energy conservation with Noether's theorem, see e.g. this Phys.SE post. – Qmechanic Oct 14 '16 at 16:21
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    I've found a video by minutephysics which supposedly does this (https://www.youtube.com/watch?v=PplaBASQ_3M), what do you think? – Suriya Oct 14 '16 at 17:48
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    I'm not sure what this question is asking for - what does deriving something "in Newtonian mechanics" but "without assuming Newton's laws" mean? The very definition of "Newtonian mechanics" is that you assume Newton's laws to be true. – ACuriousMind Oct 14 '16 at 18:17
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    @ACuriousMind that wasn't the original title, Qmechanic changed it, it originally said classical mechanics – Suriya Oct 14 '16 at 18:32
  • Well, you added "without Newton's laws" after that. Anyway, what does "classical mechanics" mean here? In Newtonian mechanics, you'd derive it from Newton's laws. In Lagrangian mechanics, you'd derive it from Noether's theorem. In Hamiltonian mechanics, it's obvious. These are the three main formulations of "classical mechanics", so what are we talking about here? – ACuriousMind Oct 14 '16 at 18:44
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    @ACuriousMind I only asked if there was any way to derive it. Why did I ask that? Because I didn't knew the answer. Had I known it wasn't possible I wouldn't have disturbed your majesty. We all don't have the honor to have a $\rho$Hd but I guess you already know that... – Suriya Oct 14 '16 at 18:51
  • I've rolled back the edit because it was inappropriate. If you've found an answer to your question, the best thing to do is post it as an answer. Don't just link to the video; give enough of an explanation of its content that the answer will count as an answer even to someone who doesn't watch the video. Also, questions and answers are not the place to comment on people's attitudes. – David Z Oct 17 '16 at 21:56
  • @DavidZ Ok so where can I complain about ACuriousMind's hateful, non-constructive, almost meaningless comments? – Suriya Oct 18 '16 at 18:54
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    Hello @SaudiBombsYemen. One of the features of the SE model is that discussions about (here) physics go on the main site, while discussions about how the site itself works belong on the meta site. You can also try participating in [chat]. Or you could try taking a walk outside in the sunshine and see if you are less bothered after a few hours of doing something different. I'll come back here later and clean up these comments. Cheers, – rob Oct 18 '16 at 19:56
  • @SaudiBombsYemen Your video assumes Newtonian mechanics. Rewatch it. At one step, he says $F=ma$. – Jahan Claes Oct 18 '16 at 20:15
  • @SaudiBombsYemen (3 comments up) When you use terminology like that, probably nobody is going to take your complaints seriously. But in general, if you see something inappropriate, flag it for moderator attention, and if you want to escalate, you can use the contact page linked at the bottom of this site to email the Stack Exchange team. – David Z Oct 19 '16 at 03:55
  • @JahanClaes thanks I hadn't realize that. Btw thanks for not responding like an asshole, I appreciate that. – Suriya Oct 19 '16 at 15:59
  • @rob I don't get what your comment has to do with nothing said here but hey thanks I guess – Suriya Oct 19 '16 at 16:01
  • @DavidZ 1. What do you mean by terminology? I asked if it was possible. Easy binary answer: yes or no. 2. I flagged the comment but it was rejected. 3. Thanks for the info although for now I'm not going to give more importance to this. Anyway thanks for answering what I asked you. – Suriya Oct 19 '16 at 16:06
  • @SaudiBombsYemen "hateful, non-constructive, almost meaningless" – David Z Oct 19 '16 at 18:38

2 Answers2

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There is no such thing as classical mechanics in the absence of Newton's laws. So the answer to that part of your question is no. To recap those laws are:

  1. Inertia (an object at rest stays at rest and an object in motion moves at a constant velocity) (strictly speaking, the first law is implied mathematically by the second law).
  2. F=Ma. Acceleration is defined as the first derivative of velocity. (Alternately, an integral of acceleration is equal to velocity subject to an integration constant that disappears in any definite integral.)
  3. Every action has an equal and opposite reaction. (Not really necessary for this analysis but included for completeness - this is a conservation of energy statement itself for the simplified case where all energy remains kinetic and is not converted to another form of energy.)

Kinetic energy is defined as work needed to accelerate a body of a given mass from rest to its stated velocity. And, work is defined as the dot product of a force vector and an infinitesimal distance vector. This in turn implies that kinetic energy which is K=(M*V^2)/2. https://en.wikipedia.org/wiki/Kinetic_energy

A conservative force (i.e. a force that conserves energy) is a force in which the energy required to go from point A to point B is independent of the path taken. https://en.wikipedia.org/wiki/Conservative_force

There are several mathematical ways that this can be shown, the most elementary of which mathematically that is truly rigorous is that a force is conservative if force is the negative gradient of potential energy (a gradient is basically a vector valued first derivative).

If you want to dispense with the mathematical rigor of vector calculus to limit your result to a one dimensional universe, for a slightly less universally true answer (and then argue by symmetry that vector calculus doesn't change anything), this implies that a force is conservative is the first derivative of potential energy is equal to the force involved, or if any definite integral of the force equation is equal to the energy required to go from the starting point to the end point.

Essentially, this can be shown by integrating to get kinetic energy using a definite integral over dV. If you can show that the result from X to Y and from Y to Z is independent of the value of Y, then it is a conservative force in one dimension, and you can then argue from symmetry that the result is true for any number of dimensions (or you can simply do the vector calculation with the gradient function for full rigor).

Ergo, Newtonian mechanic's conserves energy.

This is also true of Newtonian gravity which is another part of classical mechanics for which the force law is F=GMm/r^2, and the potential energy between two bodies can be derive to be U= -GMm/r (again by integrating over the force law in the general case). https://en.wikipedia.org/wiki/Potential_energy

Hence, a falling body turns its potential energy perfectly into kinetic energy and the amount of kinetic energy necessary to bring an object to the same height from the ground is the same.

ohwilleke
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  • But Lagrangian formalism doesn't need Newton's laws right? Sure one can deduce them, but it doesn't explicitly need them. – Suriya Oct 18 '16 at 19:04
  • @SaudiBombsYemen How does one deduce the correct form of the Lagrangian without Newton's laws? (In particular, how does one eliminate the possibility of a $v^4$ term in the Lagrangian?) – Jahan Claes Oct 18 '16 at 20:10
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    A proper Lagrangian formalism could imply Newton's laws. There are all sorts of formulations equivalent to them, but a rose by any other name . . . – ohwilleke Oct 18 '16 at 22:20
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One can prove energy is conserved with no reference to noether's theorem though in complete analogy.

Form the energy function: $E = \sum_i \frac{dL}{d(\frac{dx_i}{dt})}\frac{dx_i}{dt} - L$ where $L = L(x_{i}, \frac{dx_i}{dt}, t)$ is the lagrangian of the system.

If you take the time derivative of the energy function and use the chain rule and lagrange's equation, you will find that if L does not explicitly depend on time, that is $L = L(x_{i}, \frac{dx_i}{dt})$, with only implicit dependce on t through x and its derivative, then E will be time independent and thus conserved.

Note the equivalence to noether's theorem though not exactly the same.

CStarAlgebra
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