In the 4th paragraph he describes “the arc of a stone falling to earth” as an exponential function. Is that true? I always thought it was quadratic.
Indeed, the solution is y(t) = exp(A t) (y0 + int_0^t exp(-As) b ds).
In Einstein's theory, we have a generalization which says a stone's path through space-time follows a geodesic of the metric. In differential geometry, the exponential function, by definition, sends tangent vectors to geodesic curves.
So the author is right, under this interpretation.
Huh?
You can say “state space,” invoke the general solution to inhomogeneous ODEs, and write it out with integrals and matrix exponentials, and the answer is still quadratic. Unless you consider t^n e^(0t) to be exponential. (Hint: what are the eigenvalues of A? One could start by calculating A^2.)
Which leads to one of the most important lessons from all of physics: you can take a problem, use a different technique to solve it, and you get the same answer! It’s magic.
Maybe you would personally say that neither of these cases is truly or essentially exponential, since we have more recognizable closed forms. But then you should also commit to saying exp(it) = cos(t) + i sin(t) is not truly exponential. I would find that a little strange, but to each his own.
Anyway, I agree that you get the same answer in the end, as you must.