Recently, I'm doing my personal task which is to formalize every definition and concept in physics, by means of formal language and of course with intuitional notes. Because I found myself that the most of the concepts I'm possesing were quite informal and unclear, for instance, energy. I think I found and read many definitions about energy, but it's still a hard concept to grab.
I know what the work is. My understanding is, roughly speaking, a force $F$ is said to do work on an object $a$ when the displacement $s$ of $a$ is changed in the direction of $F$. And the amount of work $W$ is defined by $\int_C F\cdot\mathrm{d}s$ where $C$ is the path of $a$ in which $F$ is exerting. It's unit called Joule, denoted by $\mathrm{J}$. (Actually, I defined this sorely in formal language, but it is really long to write all of them and hard to read.)
And my understanding of the term energy is that the energy of an object is the highest amount of work which the object can provide to other objects.
My question is, is there any problem to understand the concept of energy like the above? Is the definition inconsistent?
Answer
This kind of definition is actually very conventional at the introductory level. You can find versions of it in many first-year textbooks. Indeed I used my version of this recipe in a recent answer.
You can probably get through an entire undergraduate education in physics on the basis of that kind of definition, but it you delve deeply enough you will eventually find it supplanted by the notion that energy is the Noetherian current associated with invariance in time.
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