system
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human
Are there cubic energy formulas?

Like potential energy is m*g*h, a linear function, movement energy is 0.5mv^2, a squared function.

Is there a function for energy that grows faster? Maybe  a cubic or an exponential one?
gpt
Yes, many. You don't see them often as textbooks like to pick simple examples.

Here's an example:

Potential energy of a charge distribution in a dielectric material is:

U = 1/2  Integral\[ rho (x,y,z) \* phi (x,y,z) dV\]

where rho (x,y,z) is the net charge density at each point (units C m^(-3)), phi  is the electric potential at each point (units V) and the integral is over the whole volume (so dV = dx \* dy \* dz and it is a triple integral).

The two functions rho (x,y,z) and phi (x,y,z) can be any function depending on your system in question.

Let say you have a box with area A in dimensions y and z and the functions only vary along X with a length of L. with rho (x,y,z) = e^(Bx) and phi (x,y,z) = Ce^(Dx) with A B C and D being constants

Then the electrostatic potential energy would be

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Integral\[ e^(Bx) Ce^(Dx) dx dy dz\] = A\*C / (B + D)  e^(Bx)e^(Dx))  evaluated from 0 to L

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or A\*C \* (e^(B\*L)e^(D\*L) \- 1) /  (B + D)

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which has an exponential dependence on the length L

**You can probably see why this does not show up in intro textbooks often.**

Edit: intro physics textbooks (like 101 and 102). As the comments pointed out, that equation is in every intro E&M textbook.