newtonian mechanics - Newton's principle of determinacy

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

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.