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

Question

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?

Answer 1

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 $\frac12 cx dx$

Answer 2

Most of them (all of your examples except $E=c^2m$, which is really just $E=m$ anyway) arise from integrating a linear equation like $p=mv$ as $E=\int v\,dp$, and it is often just a convention that we choose the linear relation to have a constant of proportionality of 1, so the integral has a constant of 1/2 (for example, we could've instead chosen, like we do with areas of circles, to have $c=2\pi r$ and $A=\pi r^2$).

Answer 3

The Kinetic Energy formula $\frac12 g(v,v) $ is fairly natural, because the motion it generates (via Euler-Lagrange equations) is geodesic motion, or "free motion." (I absorbed $m$ into the metric, $g$.) And I believe that's fully general; it would hold taking any Riemannian manifold as configuration space. I looked for a reference, and found this (page 119): http://wwwf.imperial.ac.uk/~dholm/classnotes/HolmPart1-GM.pdf

The same can be said about angular kinetic energy: free rotation is geodesic motion on the manifold $SO(3)$, if you use your kinetic energy formula as a metric. (I have the same reference for this comment.)

So the general pattern (for the first two formula, anyways) is that they are quadratic forms because they are secretly metrics on the appropriate configuration space, their use justified because they are the metrics with respect to which free motion is geodesic motion.

The others, I can't see if they fit into this pattern.