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

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.

Answer 1

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 $\frac{1}{2}m\dot x^2$.