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I am new to theoretical physics. When I was reading the proof of conservation of energy, I found that the proof was a little bit trivial and everything seems to just be a definition.

The potential energy is derived from $W=\int_c {\bf F} \, \mathrm{d}{\bf r} =\varphi(b) - \varphi(a)$ where curve $c$ starts at $a$ and ends at $b$, and $\varphi$ is the "antiderivative" of the force field ${\bf F}$, given the field is conservative. The potential energy is defined as $U(x) = -\varphi(x)$.

The kinetic energy (Newtonian), by Newton's first law, is defined as $W=\int_0^t {\bf F} \, \mathrm{d}x=\int_0^t m{\bf v} \, \mathrm{d}v=\frac{1}{2}mv^2$.

Note that in the definition, both kinetic and potential energy are derived from the definition of work, but they're just viewed differently. One is viewed as a line integral, the other is viewed as a function of velocity. However, since both are derived from work simply with a different sign, their sum is 0 for sure, I didn't see any need to do a proof. Why is a law of universe just a definition? I think it should be based on observations, but the "proof" has really made me confused, because physics is not like math and we can't "invent" laws.

Ca Parvulus Lee
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  • We aren't mathematically defining energy conservation. We are mathematically defining a quantity that we know is physically conserved (total energy). Just like how in collisions we can define a quantity called momentum that is conserved in the collision. Momentum is mathematically defined. The conservation of this quantity is physically true. – BioPhysicist Jun 26 '18 at 12:05
  • And I am not sure this is a complete "proof" of energy conservation, since not all forces are conservative with a definable potential energy, and not all objects start at 0 velocity. You are just showing that the change in kinetic energy of an object being acted upon by a conservative force can be viewed as the object losing potential energy. – BioPhysicist Jun 26 '18 at 12:10
  • yeah i didn't write down the proof. the proof assumes the force field is conservative as I mentioned in the definition, and as for the initial velocity the integral is linear so it's not a problem. – Ca Parvulus Lee Jun 26 '18 at 12:30
  • i'm now confused about "physical laws". do we first know a law and then define quantities that are consistent with the law or we first define quantities and then invent the law like theorem in math? i'm a math major interested in physics but have no idea how physics really works – Ca Parvulus Lee Jun 26 '18 at 12:32
  • With Noether's theorem, I think it's a bit more than just a definition backed by experimental observation, though, if I'm not mistaken, it did originate with some insubstantial justification. I would think that the present justification is pretty solid, and far from trivial. –  Jun 26 '18 at 12:39
  • @Chair yes this is true. Noether's theorem does seem to give conservation laws a deeper meaning beyond definitions and experiments. I was just staying at a basic level trying to separate out what we can define vs. what we see is physically true. – BioPhysicist Jun 26 '18 at 12:49
  • @CaParvulusLee I think both are present in physics. Of course physical laws must be consistent both mathematically and with our physical observations, but they must also be useful and aid in our understanding of how the world works if it is going to have any importance. We can get to that from many different ways. – BioPhysicist Jun 26 '18 at 12:52
  • @sammygerbil well not necessarily, i was just wondering if the proof really makes any sense because it looks trivial from definitions. after looking at Noether's theorem i could get the idea of why it's not trivial mathematically in that sense, though the proof given by definition of work is still kinda trivial, because they're just one thing with 2 different views – Ca Parvulus Lee Jun 26 '18 at 13:01
  • If the question is "Is the proof trivial", it becomes opinion-based. Some people will trust only solid math, while some will require an intuitive/logical understanding. That's just personal opinion, and you can't say one is universally 'more acceptable'. So if the answers in the link sammygerbil posted don't solve it, it's asking for opinions about how much the community 'trusts' present theories, which makes it off topic, as far as I understand. –  Jun 26 '18 at 13:31
  • @Chair tks for clarification, i think now i got the idea. – Ca Parvulus Lee Jun 26 '18 at 13:33
  • Just a side note, energy is not conserved in General Relativity. – user3408085 Jun 26 '18 at 15:19
  • @user3408085 it is if there's a timelike Killing vector, though I agree that in general this isn't the case. – John Rennie Jun 26 '18 at 15:20
  • No your "proof" does not make sense. It is not a proof. All that you have done is shown how to convert work into 2 different kinds of energy (potential and kinetic). That is like using exchange rates to convert dollars into euros and yen. It doesn't prove that " value" is conserved. To do that you have to go out and do some shopping. Or exchange back into dollars by some other route. To prove that energy is conserved you need to do an experiment. – sammy gerbil Jun 26 '18 at 19:13
  • @sammygerbil you're right, because i now realized the energy is defined to be "conserved" because it was proved by experiments. here it gives a very good answer regarding stuff defined in theoretical physics: https://physics.stackexchange.com/q/9686/ – Ca Parvulus Lee Jun 27 '18 at 06:34
  • @Chair I read the proof of Noether's theorem and it looks like it uses the "least action principle", which in turn assumes "conservative forces field" if you want it to be equivalent to Newton's second law. So does Noether's theorem only apply when dealing with conservative forces (given the definition of potential and kinetic energy, I think the answer is yes, otherwise they can't be equal no matter how symmetric the Lagrangian can be). – Ca Parvulus Lee Jul 03 '18 at 07:44

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