Why forces are functions of time, position and velocity at most? I know that there is a quantity called jerk (USA) or jolt (UK) which is the third-order derivative of position (i.e. the first derivative of acceleration). When we write down the second law of Newton, we write in the following generic form:
\begin{equation}
\sum\mathbf{F}(\mathbf{r}, \mathbf{\dot{{r}}},t) = m\ddot{\mathbf{r}}
\end{equation}
I am wondering what will be the consequences if we accept that there are forces acted upon the material point that depend upon higher derivatives of position. Will the initial value problem be well-defined? Can we solve it analytically? Thanks in advance.
 A: 
I am wondering what will be the consequences if we accept that there are forces acted upon the material point that depend upon higher derivatives of position. Will the initial value problem be well-defined? Can we solve it analytically?


First recapping what is True! (in certain regime),According to Newton's principle of determinacy all motions of a system are uniquely  determined by their initial positions ($\mathbf{x}(t_0)\in \mathcal{R}^N$) and initial velocities  ($\dot{\mathbf{x}}(t_0)\in \mathcal{R}^N$).
There is a function $\mathbf{F}:\mathcal{R}^N\times\mathcal{R}^N\times\mathcal{R}\rightarrow \mathcal{R}^N$ such that
$$\ddot{\mathbf{x}}=\mathbf{F}(\mathbf{x},\mathbf{\dot{x}},t)$$
It's called Newton's equation.
By the theorem of existence and uniqueness of solutions to ordinary differential equations, the function $\mathbf{F}$ and intial conditions uniquely determine a motion.

Galileo's principle and  Newton's differential equation are basic experimental facts  which lie at foundation of mechanics.
Now suppose You found (Very nice example is Dirac-Lorentz equation) that force depend on higher derivative so that Newton's differential equation now look like
$$\mathbf{F}=\mathbf{F}(\mathbf{x},\mathbf{\dot{x}},\mathbf{\ddot{x}},\cdots,t)$$
Then by the theorem of existence and uniqueness (Picard–Lindelöf theorem) of solutions to ordinary differential equations, the function $\mathbf{F}$ and initial conditions $\mathbf{x}(t_0)$ $\dot{\mathbf{x}}(t_0)$, $\ddot{\mathbf{x}}(t_0)$ upto ($n-1$)th derivative of position.

Might be interested :
Is there any case in physics where the equations of motion depend on high time derivatives of the position?
Why are there only derivatives to the first order in the Lagrangian?
