Why is dynamics first order in phase space?

Question

I have watched some lectures in which the lecturer said that system dynamics are (generally?) first order in phase space, forming a system of coupled differential equations. At a basic level I see this as reflecting Newton's second law $\frac{\partial ^2 x}{\partial t^2}=\frac{\partial p}{\partial t}/m$, however it is not at all clear to me that this is general (if it is).

Answer 1

1. A coupled system of DEs with higher time-derivatives can be converted to a system of DEs with only first-order time derivatives by introducing more variables.

2. A Hamiltonian formulation in a phase space is first order (i.e. contains no time derivatives of more than first order) by definition.

3. See also this & this related Phys.SE posts.