# Show That A Particle In A Bounded Force Field Can Reach Any Point In Fixed Time Span

I tried to proof 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 $B_r\subseteq C$.

• 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.
– Neal
Jun 24 '19 at 22:07