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I tried to prove that for a smooth bounded force field $F$ and $x\in{\bf R}^n$ there exists some $v\in{\bf R}^n$ such that a particle starting in $0$ with mass $1$ and velocity $v$, obeying Newton's second law, will reach $x$ after exactly $t$ time units for some $t>0$. By rescaling impulse and force field, we can w.l.o.g. assume $t=1$.

Define $r:=\Vert x\Vert+1$ and let $\phi_G(v,.)$ denote the flow of such a particle starting with velocity $v$ in a force field $G$. Given that $F$ is bounded, there is some constant $c$ with $$\Vert\phi_{\alpha F}(v,1)-v\Vert<c\tag{1}\label{1}$$ for all $\alpha\in[0,1],v\in{\bf R}^n$. I believe, we can show $B_r\subseteq\phi_F(B_{r+c},1)$.

I'd like to do something like this: First, the degree of the map $\phi_F(.,1)$ on $\partial B_{r+c}$ is $1$. This should follow from $(1)$ which implies that $(u,\alpha)\mapsto\phi_{\alpha F}(u,1)$ is a homotopy between the identity on $\partial B_{r+c}$ and $\phi_F(.,1)$ which never maps to $0$. Now infer (with some result from algebraic topology I reckon) that $\phi_F(B_{r+c},1)$ contains the connected component $C$ of ${\bf R}^n\setminus\phi_F(\partial B_{r+c},1)$ which contains $0$. Using $(1)$ again we have $B_r\subseteq C$.

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  • $\begingroup$ I wonder if you could pick $v = C|x|$ for sufficiently large $C$ and then use continuity of solutions of ODEs with respect to initial conditions. $\endgroup$
    – Neal
    Commented Jun 24, 2019 at 22:07
  • $\begingroup$ How does the time appear in the logical quantification? Is it $\forall F\,\forall x\,\exists v\,\exists t$, or is it $\forall t\,\forall F\,\forall x\,\exists v$, or something else? $\endgroup$
    – mr_e_man
    Commented Dec 29, 2022 at 1:30

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I'd say this definitely depends on the force field $F$. Take for example a particle at position $r(t)=0$, and say $V_r(0)=1$ (this is the particle's initial velocity, and $V_r$ is the particle's velocity). Next choose $F=1/r^2$ (add in the proper units). There exists a set of points which are unreachable by the particle, because after a finite time the velocity will become negative, and oscillate forever. But this $F$ above is unbounded. However, I could just as easily make a bounded force field with a step function and apply the same analysis as long as the force field was present for all of the particle's trajectory; there may be a more general statement here though.

So I would say that a particle with a given initial velocity cannot necessarily reach a given point in a given bounded force field in a fixed time span.

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    $\begingroup$ What is $V_r$ ? $\endgroup$
    – mr_e_man
    Commented Dec 29, 2022 at 0:10
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    $\begingroup$ For a given initial velocity, there are some unreachable points. But if the initial velocity is allowed to vary, then any point can be reached. $\endgroup$
    – mr_e_man
    Commented Dec 29, 2022 at 0:21

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