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| ▲ | drivebyhooting a day ago | parent | next [-] |
| Energy conservation comes directly from integrating Newton’s second law one time, assuming a conservative force field: 1. F = ma 2. -dV/dx = m d2x/dt2 # force is the negative gradient of potential. Acceleration is the second time derivative of displacement. 3. Rewrite d/dx as 1/v * d/dt via the chain rule: d/dt V = d/dx V * dx/dt => d/dx V = 1/v d/dt V => d/dx = 1/v d/dt. 4. Rearrange (2).
0 = dV/dt + m v d2x/dt2. 5. Integrate both sides by t.
E = V + 1/2 m v^2 # where the constant of integration is a conserved quantity (energy). |
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| ▲ | saghm a day ago | parent [-] | | You've already lost me by step two; I have no clue what "force is the negative gradient of potential" means or what a "conservation force field" is. When my non-expert perception is that potential energy already seems a bit like a hack to cause conservation to work, citing some technical fact based on the assumption that there's something called "potential" with a negative gradient that I should take for granted as equivalent to force doesn't really do much to dispel that. It seems like your response is assuming I understand a lot more than I do, and I suspect that if I understood enough to do anything other than just trust it at face value, I probably wouldn't need any explanation about why my intuition is wrong in the first place! Circling back to the blog post here, it seems like the author is specifically trying to discuss things in a way that non-experts can still follow along with I took a lot at a couple of other posts on the blog after finishing this one, and the one about a "proof" that pi equals 4 had a section that felt pretty similar to me, where it cites an explanation of why it's wrong conveyed in as way that probably doesn't do much to help the people who are most likely to need it in understanding why the proof is wrong: https://lcamtuf.substack.com/p/4 |
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| ▲ | zokier a day ago | parent | prev | next [-] |
| > we just came to with a convenient way to do calculations, slapped a name on it, and declared it a scientific law Physics is a model after all. I don't think you are so far off. Lot of physics is predicated on it's ability to make predictions, and predictions often come in a form of calculation of something. There is even a saying "shut up and calculate" in physics. It is not without contention, but it does describe a lot of physics especially today. It is also notable that you can have multiple different models about the same thing. Notoriously we have both lagrangian and newtonian mechanics, both which are valid but can be useful in different scenarios. |
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| ▲ | BrandoElFollito a day ago | parent | prev | next [-] |
| Well, physics is primarily an experimental science. We see something, we make a model that hopefully brings something new to check, we check, and either it is a good model or it is not. Then come the theoreticians that try to find something deeper that would link or emerge the models. In the case of energy conservation the "root" reason for the conservation are Noether principles that state some immutable entities (basically an experiment is not dependent on time and on geometric transformations such as translation or rotation). The link is not immediately visible (5 pages of math :)) but then suddenly tadaaam! and conservation of energy! But yes, your point is very correct and this is why people wondered (and to some extend still wonder) if the masses in mhg=mv^2/2 are the same masses :) Physics is really cool |
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| ▲ | lanstin 19 hours ago | parent | prev [-] |
| Conserved quantities arise when there is a symmetry in the laws of physics - energy conservation in Newtonian mechanics is basically a reflection of the observed fact that the laws of physics are the same in the past, present and (and one guesses) future. The way (in a mechanical system) for energy to be created would be if things sped up or slowed down more than Newton's laws allowed. |
