Coincidence, purposeful definition, or something else in formulas for energy

In the small amount of physics that I have learned thus far, there seems to be a (possibly superficial pattern) that I have been wondering about.

The formula for the kinetic energy of a moving particle is $\frac{1}{2}mv^2$.

The formula for kinetic rotational energy is $\frac{1}{2}I\omega^2$.

The formula for energy stored in a capacitor is $\frac{1}{2}C \Delta V^2$.

The formula for energy delivered to an inductor is $\frac{1}{2}LI^2$.

Finally, everyone is aware of Einstein's famous formula $e=mc^2$.

I realize there are other energy formulas (gravitational potential energy, for example) that do not take this form, but is there some underlying reason why the formulas above take a similar form? Is it a coincidence? Or is there a motivation for physicists and textbook authors to present these formulas the way they do?

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+1 for [tag: intuition] questions even after the tag was murdered. – centralcharge Jul 15 '13 at 4:25
intuition, I mean. – centralcharge Jul 15 '13 at 9:44

Usually, linear equations are very common in physics. Something like $\text{Quantity}=\text{constant}\times\text{variable}$

So, you have $p=mv,L=I\omega,Q=CV$, etc.

Now, it just turns out that when you multiply these linear equations with a small increment of the variable, you get an expression for energy. Why this happens isn't so easy to figure out. The easiest explanation is that we usually define our variables such that force and similar quantities (e.g, net charge, etc) are linear.

Now, if you sum up a linear equation multiplied by a small change in the linear part, you get something like $\int cx dx$, which is a quadratinc term of the form $\frac12 cx dx$

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Thanks - I understand now that my question was basically silly :) – tacos_tacos_tacos Jul 27 '13 at 22:52

Not exactly a coincedence. These kinds of equations:

$$p=mv$$

Are very very very very very very simple and are clearly quite common (perhaps definitions, or proportionalities, or etc.). But if you integrate this equation with respect t $v$, then

$$K=\int p\mbox{ d}v=\frac12mv^2$$

Which takes the form you mentioned,. It' is often the same with other such equations, including $\pi r^2$ (P.S. $\pi$ reminds, me don't support the tau manifesto).

As for Mass-Energy Equivalence, well, the proof is quite different.

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