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?

on$\mathbb{R}^3$ can have, but no solution of a second order ODEon$\mathbb{R}^3$, i.e. on thesamespace. – student Oct 23 '10 at 16:34