David, Enon, Daniel, John and all, Thank you for this question which I hope to absorb along with all of your deep answers.
As regards intuition, I am thinking that another way to say what has been said here is:
Nature minimizes kinetic energy, but relative to what? Relative to the potential energy. So the potential energy is just the baseline, the "zero", the absolute, and kinetic energy is what is relative to that. That's why we subtract it.
Note that potential energy is typically infinite, as in the case of an object's gravitational field or electromagnetic field. Whereas kinetic energy is definitely finite.
So we have to restrict ourselves to considering a finite range of potential energy.
And within that range we consider the exerted action which is also finitely delineated. And the action is the (integrated) sum of what the force exerts over space and time.
I am picking up from Dan Piponi's answer at Quora
this distinction between the "bottom-up" (building up a space with vectors) and "top-down" (tearing down a space in terms of hyperplanes/reflections/covectors) which keeps coming up as I try to understand tensors. So it seems perhaps that kinectic energy is the bottom-up view of energy (going up as the space is built up) and potential energy is the top-down view of energy (going down as the space is broken down).
I've also been trying to understand the intuitive difference amongst the classical Lie groups, the unitary, orthogonal and symplectic groups. And it seems to boil down to how to "undo" an action:
* Unitary: conjugate transpose
* Orthogonal: symmetric transpose
* Symplectic: anti-symmetric transpose
And the transpose has to do with switching from vectors to covectors in their duality. Given this commonality, then there are three "easy" ways to invert the action, the matrix. And I think the three have to do with ways of thinking about the relationship between the complexes and the reals, and in particular, the philosophical relationship between unmarked opposites (like the two solutions to the square root of -1, call them i and j) and marked opposites (like i and -i). But I'm just imagining as I try to understand.