How did people come up with various expressions for energy? By this I mean the expression or formulas for kinetic energy, gravitational potential energy etc. and the intuitive guiding principles to arrive at them.
Energy seems to be too abstract a quantity and with also a very loose definition. If energy was an abstract quantity which when calculated, always remained constant and no one really knows what it is, as Feynman puts it, how do you come up with mathematical expressions for it.
I guess a better way would to put it would be, that if you have identified a form of energy, how do you come up with a way of calculating it ?
Also, I would also like to know why work is defined as force times distance and how is this definition of work equivalent to change in the energy of a system. A historical and intuitive explanation would be appreciated.
PS: I am not just asking why work is defined to be force times distance, but how overall work is considered to be a change in the energy of a system and how this fits in the complete picture of energy as an abstract quantity.
Frankly speaking if someone can just shed light on this whole energy thing, it would be helpful as I am having nightmares trying to grasp this.