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

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 $$\Vert\phi_{\alpha F}(v,1)-v\Vert for all $$\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
Commented Jun 24, 2019 at 22:07
• 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? Commented Dec 29, 2022 at 1:30

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.
• What is $V_r$ ? Commented Dec 29, 2022 at 0:10