TITLE: Newton equations, second order equation and (im)possible motions
QUESTION [15 upvotes]: I am am currently studying Newtonian mechanics from a conceptional and axiomatic point of view. Now, if I am not mistaken, one (but surely not all) statement of Newtons second law about nature is, that (almost ?) every motion of a "classical particle" (or "small body") can be described by a second order differential equation on $\mathbb{R}^3$ (or $\mathbb{R}^{3n}$ if one considers a system of $n$ particles). That is, if $I \ni t \mapsto x(t) \in \mathbb{R}^{3n}$ is a motion of $n$ particles in some environment, there is a smooth function $f \colon \mathbb{R}^{3n} \times \mathbb{R}^{3n} \times \mathbb{R} \to \mathbb{R}^{3n}$ (which describes the influence of the environment) such that $\ddot x(t) = f(x(t),\dot x(t), t)$ for all $t \in I$, thereby $I$ is an interval and $\dot x$ denotes the derivative of $x$. 
Now my question is, if there is a good, mathematical sound intuition, which kinds motions are not allowed by Newtons second law because of the fact that it is a second order differential equation.
In other words: I want to analyze in detail, what Newtons seconds law tells us about nature. Especially I want to grasp, how the condition to be a second order differential equation gives restrictions to possible conceivable motions of particles. What would be allowed additionally if the equations were of $3$. or higher order?

REPLY [2 votes]: This is maybe  a bit of a simpleton's answer:
Motions of particles are described by second order equations because two particles starting from the same position and with the same velocity follow the same trajectory. 
At least no experiment has ever shown evidence of the contrary.