Suppose we are given an ODE of a system, for example, a mass spring damper:
$$m\ddot q + \dot q + q = u$$
We may think of $q$ as position, $\dot q$ as the velocity, $\ddot q$ as the acceleration.
Why is it that we only assign states to $q$ and $\dot q$, i.e., letting $x_1 = q$ and $x_2 = \dot q$, instead of letting letting $x_1 = q$ and $x_2 = \dot q$ and $x_3 = \ddot q$?
Is it not that the acceleration is a "state" of the system? If not, then why should the other two variables be the state?
I'm watching a video on Youtube where they attempt to assign state variables to a swinging pendulum, and the video says, in which I paraphrase:
"since kinetic energy and potential energy depends on position and velocity, therefore it makes sense to assign these as state variables."
I'm looking for a more precise way of stating the above quote.