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

Question

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 $B_r\subseteq C$.

Answer 1

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.