Why does it happen that $mv^2/2$ - kinetic energy - is quantity associated with movement that transforms into other forms of energy?

Question

I'm having a hard time figuring out why it is that, of all possible scalar quantities associated with the movement of things, $\frac{mv^2}{2}$ is the one that neatly wraps up conservation of energy into a cohesive conversion of energy statement applicable to physical systems of any mixed type, mechanical-chemical, mechanical-heat, mechanical-electromagnetic, and so on - and not only mechanical energy in a purely mechanical system.

We can derive the work-energy theorem from Newton's Second Law and very directly apply the kinetic energy formula to a system whose constituents interact mechanically, no question there. But how come it turns out that it is that particular quantity that also encapsulates how systems with interactions other than mechanical might behave? An example: we know that an object that comes to a stop without deformation when hitting another object and raises its temperature by $\Delta T$ will raise that temperature by $4\Delta T$ if we double that object's speed.

There is a really brilliant answer to a similar question to this by Ron Maimon from back when he used to be a regular here: https://physics.stackexchange.com/q/535/ — doublefelix, Dec 06 '22 at 22:15
@doublefelix Thank you for that link, it was a great read! I guess the next step in the "why" chain is to ask why moving frames don't measure different temperatures in that experiment. Are there more fundamental principles in action? — jvf, Dec 06 '22 at 22:28
Does this answer your question? Why does kinetic energy increase quadratically, not linearly, with speed? — Farcher, Dec 06 '22 at 23:21

score 1 · Answer 1 · answered Dec 07 '22 at 02:59

If you start with Newton’s 2nd law you can derive the Euler Lagrange equations with the Lagrangian $$\mathcal{L}=\frac{1}{2}m\dot x^2-V(x)$$

Since this Lagrangian is time independent, Noether’s theorem tells us that there is a conserved energy and allows us to derive the form as $$\mathcal{H}=\frac{\partial \mathcal{L}}{\partial \dot x}\dot x -\mathcal{L}=\frac{1}{2}m\dot x^2 + V(x)$$

When we go beyond Newton’s laws and start constructing Lagrangians that produce other equations of motion we find that those Lagrangians still are time independent and that the resulting Noether conserved quantity has a term like $\frac{1}{2}m\dot x^2$.

Why does it happen that $mv^2/2$ - kinetic energy - is quantity associated with movement that transforms into other forms of energy?

1 Answers1