Newton's principle of determinacy

Question

I am a mathematician. I have a (somewhat long term) goal of understanding some of the physical insights that have influenced my area of research. To this end I read Arnold's Mathematical methods in classical mechanics a while ago, but something I didn't understand has been bugging me ever since.

In the first chapter Arnold defines a motion of $n$ particles in $\mathbb{R}^3$ as a map $\mathbf{x}:\mathbb{R} \rightarrow \mathbb{R}^N$ for $N=3n$. The first chapter is then about what types of motion are allowed. In section 2D Arnold makes the following observation:

According to Newton's principle of determinacy all motions of a system are uniquely determined by their initial positions ($\mathbf{x}(t_0) \in \mathbb{R}^N$) and initial velocities ($\mathbf{\dot{x}}(t_0) \in \mathbb{R}^N$).

This seems important and I expect that this would have an impact on what types of functions $\mathbf{x}$ could be. He goes on:

In particular, the initial positions and velocities determine the acceleration. In other words, there is a function $\mathbf{F} : \mathbb{R}^N \times \mathbb{R}^N \times \mathbb{R} \rightarrow \mathbb{R}^N$ such that $$ \mathbf{\ddot{x}} = \mathbf{F}(\mathbf{x},\mathbf{\dot{x}},t). $$

So the implication of Newton's determinacy principle is that the acceleration obeys a second order differential equation. This seems completely vacuous to me. Any function $\mathbf{x}$ obeys a second order differential equation (as long as it is twice differentiable).

Could someone please explain to me what Arnold is saying here. I feel like I am missing something important.

Answer 1

The trajectories are uniquely determined means that the theorem of existence and uniqueness applies (so, the differential equation has to be sufficiently regular).

Newton's principle states more: the system is fully determined by the position and the speed, that is, by $2n$ constants, where $n$ is the dimension of the space. As you have $n$ equations (one per spacial coordinate), they are completely determined if and only if they are of second order.

The statement is not that the function $x$ obeys a second order differential equation, it says that the dynamics are directed by a second order DE.

Edit:

In other words, The key is that there is one set of ODE for any possible initial condition. You can construct a first order ODE for a given trajectory, but it will be useless if you change the initial conditions.

Answer 2

A very simple motivation for writing $\ddot{\bf x}(t) = {\bf F}({\bf x}, \dot{\bf x}, t)$, which might shed some light, is the following. We are given ${\bf x}(t)$ and $\dot{\bf x}(t)$ and we desire to calculate ${\bf x}(t + \delta t)$ and $\dot{\bf x}(t + \delta t)$. Now, ${\bf x}(t + \delta t) = {\bf x}(t) + \dot{\bf x}(t) \delta t$, which we may calculate. But $\dot{\bf x}(t + \delta t) = \dot{\bf x}(t) + \ddot{\bf x}(t) \delta t$, which in order to calculate we require to know $\ddot{\bf x}(t)$. By Newton's law of determinacy, the motion is calculable given ${\bf x}(t)$ and $\dot{\bf x}(t)$. Thus there must be a function ${\bf F} : \mathbb{R}^N \times \mathbb{R}^N \times \mathbb{R} \rightarrow \mathbb{R}^N$ such that $\ddot{\bf x}(t) = {\bf F}({\bf x}, \dot{\bf x}, t)$. What this function is constitutes a law of nature.

Answer 3

What I understand from the qualitative statement is that of all possible laws that acceleration could "obey," it actually obeys a 2nd ODE! It is irrelevant, whether or not, other functions also obey the same law.