The question I will ask makes sense in much more generality, but I will leave the translation to the experts, since I'm only looking for a special case (and it would not surprise me if the answer does not generalize). I will give some background, and then ask my question as a conjecture, set apart from the main text.


Let $\mathbb R^n$ have its usual metric, and pick a differential one-form (= vector field) $B$ (the "magnetic potential") and a differential zero-form (= function) $C$ (the "electric potential"). Then consider the following second-order ODE for parameterized paths $\gamma: [0,T] \to \mathbb R^n$:
$$ 0 = \ddot \gamma + dB \cdot \dot\gamma + dC \quad\quad \text{(EOM)} $$
I'll let you pick the signs for how the two-form $dB$ eats the vector $\dot\gamma$; just be consistent.


Then (EOM) is nondegenerate, and so a solution is determined by its initial conditions $(\dot\gamma(0),\gamma(0))$. For each $T \in \mathbb R$, let $\phi\_T: \mathbb R^{2n} \to \mathbb R^n$ be the "flow by time $T$" (actually, it is defined only on an open subset of $\mathbb R^{2n}$, given by $\phi\_T(v,q) = \gamma(T)$ for the solution $\gamma\\,$ to (EOM) with initial conditions $(\dot\gamma(0),\gamma(0)) = (v,q)$. Then $\phi\_T$ is smooth; in fact, it is smooth in the $T$ variable as well. This follows from some standard fundamental result in ODEs, for which I don't have a good reference.


A path $\gamma: [0,T] \to \mathbb R^n$ is *classical* if it satisfies (EOM); its *duration* is the number $T$. We can also consider paths with negative duration by flowing backwards, although we will not need to do so.


**Definition:** A point $(v,q) \in \mathbb R^{2n}$ is *focal for duration $T$* iff ($\phi\_T(v,q)$ is defined and) $\det(\partial \phi\_T(v,q)/\partial v) = 0$; i.e. fix the $q$, think of $\phi\_T(-,q)$ as a function of $v$ only, and ask that its differential is degenerate. By identifying $(v,q)$ with its classical path, we will talk about "focal (classical) paths" for given durations.


It is a standard results (see e.g. Milnor's *Morse Theory*) that for a given point $(v,q) \in \mathbb R^{2n}$, the durations $T\in \mathbb R$ for which it is focal are discretely separated. Note that every $(v,q)$ is focal for duration $T=0$.


**Proposition:** Let $\gamma$ be a classical path of duration $T$. Then it is non-focal if and only if it extends to a family of classical paths smoothly parametrized by the boundary positions $(\gamma(0),\gamma(T))$.


**Sketch of Proof:** Being focal for duration $T$ is a closed condition on $\mathbb R^{2n}$, so we can vary $\gamma(0) = q$ while remaining non-focal. But for non-focal paths we can vary $\gamma(T)$ via the inverse function theorem.


Anyway, pick $q \in \mathbb R^n$, and $v = B(q)$ (or $-B(q)$ depending on your sign convention: for experts, I want the momentum to vanish). Then for some $\epsilon>0$, for all $T\in (0,\epsilon)$, $(v,q)$ is non-focal for duration $T$. Thus, for each $T \in (0,\epsilon)$, I can find an open neighborhood $q \in \mathcal O\_0 \subseteq \mathbb R^n$ and another open neighborhood $\mathcal O\_1 \subseteq \mathbb R^n$ so that for $(q\_0,q\_1) \in \mathcal O\_0 \times \mathcal O\_1$, there is a non-focal classical path $\gamma$ of duration $T$ with $\gamma(0) = q\_0$, $\gamma(T) = q\_1$, depending smoothly on the boundary conditions, and such that the classical path of duration $T$ and initial conditions $(\dot\gamma(0),\gamma(0)) = (v,q)$ is contained within this family.


Note that as $T \to 0$, the classical path with initial conditions $(\dot\gamma(0),\gamma(0)) = (v,q)$ ends at a point very close to $q$. I don't know if I can take $\mathcal O\_1$ to actually contain $q$.


I would like to reverse the direction of choices: I'd like to pick $\mathcal O\_0,\mathcal O\_1$ first.



> 
> **Question/Conjecture:** Let $q \in \mathbb R^n$. Then there exist open neighborhood $\mathcal O\_0,\mathcal O\_1 \subseteq \mathbb R^n$, with $q \in \mathcal O\_0,\mathcal O\_1$, and $\epsilon>0$ such that:
> 
> 
> 1. There exists a family of classical paths $\gamma$ with boundary values varying in $\mathcal O\_0,\mathcal O\_1$ and with duration varying in $(0,\epsilon)$. I.e. let $\Delta = \{ (T,t) \in \mathbb R^2 : T \in (0,\epsilon), t\in [0,T] \}$; then there is a smooth function $\gamma: \mathcal O\_0 \times \mathcal O\_1 \times \Delta \to \mathbb R^n$ with: (a) $\gamma(q\_0,q\_1,T,-)$ is classical for each $(q\_0,q\_1,T) \in \mathcal O\_0 \times \mathcal O\_1 \times (0,\epsilon)$, and (b) $\gamma(q\_0,q\_1,T,0) = q\_0$ and $\gamma(q\_0,q\_1,T,T) = q\_1$.
> 2. For each $T \in (0,\epsilon)$, the classical path of duration $T$ with initial conditions $(B(q),q)$ appears as some $\gamma(q,q\_1,T,-)$.
> 
> 
> 


For comparison, the corresponding theorem about geodesics on a Riemannian manifold is standard: around any point you can find a small neighborhood such that any two points in the neighborhood can be connected by a unique geodesic that does not leave the neighborhood. In fact, it follows from the proposition and the observation that changing the duration of a geodesic for fixed boundary conditions amounts just to a linear reparameterization.



Do there exist small neighborhoods in a classical mechanical system without pairs of focal points?
For a sufficiently large particle energy, the original problem can be transformed to a problem of geodesic motion as follows:


The motion of a classical particle in an external magnetic field in n-dimensions can be 
seen as a symplectic reduction of a geodesic motion in n+1 dimensions (Rn \* S1) through the 
Kaluza-Klein construction, given for example in section 7.6 of [Marsden's book.](http://books.google.com/books?id=I2gH9ZIs-3AC&printsec=frontcover&dq=Marsden+%2B+Introduction+to+mechanics+and+symmetry&source=bl&ots=STIWna6Ib8&sig=qrkEUDWwL1mbzkoNPA8OPiAWyEY&hl=en&ei=ps5uS8KQDpWCnQPN3ZnSBA&sa=X&oi=book_result&ct=result&resnum=2&ved=0CBEQ6AEwAQ#v=onepage&q=&f=false) The remaining problem is a geodesic motion in an electric potential field. 
Now suppose that there exists a region in the vicinity of the origin where the 
electric potential is bounded from above, and the particle's energy (The value of the 
Kaluza-Klein Hamiltonian which is a constant of motion) is larger than the maximum potential.
In this case, the trajectories in this region are equivalent up to a reparametrization 
to a free geodesic motion in the Jacobi metric (see section 7.7). 
Thus, in this case, the original problem is equivalent to a Riemannian problem.


