What led to the formula $W=\vec{F} \cdot \overrightarrow{d s}$?

Question

For me the concept of energy in physics is the least intuitive one as in all the books on Physics that I have read so far. The concept of energy have been the least intuitive one as I cannot understand How these things actually came into existence.

Even the definition of Energy is a bit vague, many books define Energy as the ability to do work but then the question comes that what is work ? then they say that work is the process of transfer of energy or to make it even more boring they will just read out the formula for work done as ( except Fundamentals of Physics by Resnick Halliday which admits that it is quite difficult to write a definition of energy).

So to get a better understanding of this concept of energy I started reading about the origin of this concept of energy in Physics like the Vis Viva controversy, answers on this site related to my questions like How was the formula for kinetic energy found, and who found it? , How was the definition and the formula of work derived? and a few more material that I could get, I even tried to find the research papers related to all this couldn't find them. And by my experience of reading all this material I felt that this is not helpful to any curious student because as a student I am particularly interested in the experiments that lead to the concept of energy, the observations made by the genius minds that leads to result that we study ( for example, Leibnitz observed that $\Sigma m_{i} a_{i} = constant$ fore systems with no friction in case of elastic collision ) but these articles and answers are more focused on the history of who did what, who said what rather focusing on How did they do that ?

The timeline of all the system is very confusing ( more than the plots of Christopher Nolan's movies ) and more importantly useless to a student ( since student, it maybe interesting to a history student ).

To summarise:

What I am looking for is answer/ article/ website anything that can help me to understand How this concept of Energy works ? How are the results that we study like $$ \begin{array}{l} d W=\vec{F} \cdot \overrightarrow{d s} \\ K \cdot E=\frac{1}{2} m v^{2} \end{array} $$

came into existence, what experiments lead to these conclusions?

In a nutshell, I want to know what experiment and observations lead to the modern concept of energy I am not interested in knowing the names of the scientist rather I am interested in there experiments, observations and mathematics

I know all the above is very lengthy but it was important to prove that this is not a homework type question.

This is my question : What were the experiments and observations that lead to the formula $ W=\vec{F} \cdot \overrightarrow{d s}$, if you know any derivation (except the one using Kinetic Energy ) of this please let me know.

Hope this will clarify my question to you.

I’m voting to close this question because it is not a question about the history of science or mathematics. — kimchi lover, Oct 17 '21 at 12:31
@kimchilover WTF ? At Physics stack exchange site they say it's a HSM question and close the question and here you say it's not — user1086173, Oct 17 '21 at 12:37
"What is the most intuitive way to understand X?" is not a history question. The HSM and Physics folks reacted in different, but equally valid, ways to the mismatch between your question's title and the question's last paragraph. — kimchi lover, Oct 17 '21 at 13:47
Perhaps you should visit Philosophy.SE and learn why "ad hominem" is not a good thing. — Ray Butterworth, Oct 17 '21 at 13:53
Does this answer your question? How did people figure out the formula for mechanical work, and related it to energy? A big part of it were Boscovich's metaphysical musings. Maybe what you are looking for is some kind of "rational reconstruction" from observations and experiments, but they played little role in this case, so history cannot provide that. — Conifold, Oct 17 '21 at 20:42

sand1 · Answer 1 · 2021-10-17T17:55:59.673

Mirowski P., More heat than light (1989 Cambr.UP), Chap.2. pp.11-98 The history of the energy concept.< It's a great book for various other reasons.>

For Ernst Mach, energy was more or less defined as the ability to do work. Although "work" could mean many things to many people, Mach felt it was merely an historical accident that the term was defined mechanically in early physics: that is, lifting a known weight through a certain distance. (p.15)

So, here is the key observation, some details and derivation are given in the text of this chapter. The problem https://philosophy.stackexchange.com/questions/85899/is-the-idea-that-everything-is-energy-even-coherent/85902#85902 was recently discussed and the book mentioned there /Coopersmith J., (Oxford UP, 2010) Energy, the Subtle Concept. / has also historical details.

What led to the formula $W=\vec{F} \cdot \overrightarrow{d s}$?

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