Definition of kinetic energy without the second Law of Newton

Question

As I see it, the definition of kinetic energy $$T= {1\over2} m u^2 \text { where $u<<c$}$$ comes by using the definition of work $$W= {\int F\cdot\ dx }$$ and we use for the meaning of F(force) the Second Law of Newton: $$F={dp\over dt}=ma$$

Do I understand correctly that the kinetic energy from this point and on becomes a connection link between Newtonian mechanics, theoretical mechanics(Lagrange, Hamilton) and relativity?

If so, this the question: Can there be a definition of energy without the law of Newton?(it seems to me that to use the law of newton is not wrong but strange to the point of view that comes with theoretical mechanics. We remain somehow bounded to a definition of kinetic energy associated with a law that makes the mass something a little more fundamental from let's say charge-by fundamental I mean that something without mass, Newton says it cannot move or have any type of interaction. But if the charge is another way of interaction-via the electromagnetic fields or potentials- shouldn't we search for the possibility of defining the kinetic energy via the charge. What should become of the potentials then? And why mass in the Newton's law?).

Thank you.

PS:if anyone has a recommendation for further study, he can suggest it.

Answer 1

You can define the kinetic energy as you want. As long as they are consistent with each other, definitions are yours to choose.

For example you can take $$T = \frac{1}{2}m v^2 \, ,$$ as a definition and use it to prove Newton's law by requiring energy to be conserved, $$ \frac{dE}{dt} = \frac{dT}{dt} - \frac{dW}{dt} = m v a - F v = 0\, .$$

If you want a less silly answer, you can use the relativistic definition $$ T = \sqrt{p^2 c^2 +m^2 c^4} \, ,$$ ($c$ is the speed of light.) which stays valid for $m=0$.

About your parenthesis: It should not bother you that mass has a special place in the laws of physics as compared to electric charge for example. It is true that in classical physics the mass is the gravitational charge, but is it also (and more importantly) a measure of the inertia of things. Newton's second law, $F = m a$, tells us that it takes more force to accelerate heavy objects. This has nothing to do with gravitational interactions. See this post.