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?
Best Answer
Actually, Newton did NOT say that $F = m a$ (i.e., ${d^2 x \over dt^2} = {1\over m} F$) in the Principia. First of all, if he did, no one at that time would have understood what it meant since that was pre-calculus times. What he did say in his 2nd Law was that "Change of motion is proportional to impressed motive force and is in the same direction as the impressed force", i.e., in modern terminology, the instantaneous change of momentum (caused by something like a hammer blow) is equal to the applied impulse. Whenever he used the 2nd Law, he treated a smooth force as the limit of a large number of small impulses. It was only much later that Euler recast the 2nd Law as a 2nd order ODE. This is all discussed in considerable detail in a recent book ``Differential Equations, Mechanics, and Computation'' (that I co-authored with my son Robert). There is a "Web Companion" for the book at the URL http://ode-math.com where you can freely download more than half the book as pdf files. In particular, if you download the first pages of "Chapter 4: Newton's Equations" you will see all of this discussed in considerable detail. One more point: this book was explicitly written to be, as we stress, a conceptual introduction to the subject for someone like yourself who is learning this material for the first time. (See here: http://ode-math.com/NovelFeaturesOfODECM.html)
All of the above is somewhat tangential to your specific question, so let me add that a very major restriction imposed by saying that the laws of motion for say $n$ particles are (equivalent to) 2nd order ODE is that if you know the positions and velocities of the particles at any one instant then their positions at any time in the past and future are uniquely determined by that data. Or as Laplace said in a very famous quote "The current state of Nature is evidently a consequence of what it was in the preceding moment, and if we conceive of an intelligence that at a given moment knows the relations of all things of this Universe, it could then tell the positions, motions and effects of all of these entities at any past or future time. . ." (Of course we now know that the existence of "chaotic behavior" renders that only a very theoretical possibility.)