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# Do there exist small neighborhoods in a classical mechanical system without pairs of focal points?

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.