Are Energy Laws in Physics Really Just Coincidences?

[MHD93]

Hello

I've been wondering a lot of why many laws of energy in different branches of Physics seem similar, for example:

Kinetic:
$$\frac{1}{2}mv^{2}$$
Spring:
$$\frac{1}{2}kx^{2}$$
Rotational:
$$\frac{1}{2}I\omega^{2}$$
Capacitor:
$$\frac{1}{2}VC^{2}$$
Capacitor:
$$\frac{1}{2}eE^{2}$$
Magnetic:
$$\frac{1}{2}LI^{2}$$

Does this have a real physical interpretation, or is it is just by accident?

 

[Disinterred]

A question I have been wondering about too.

Not too important here but, I believe (1/2)Iw^2 was derived using (1/2)mv^2, so that connects two of them.

 

[russ_watters]

If the units are the same the math has to be similar.