[drivebyhooting]

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).

[saghm]

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