### Background

Lagrangian mechanics on $\mathbb R^n$ is usually defined by picking a **Lagrangian** function $L: {\rm T}\mathbb R^n \to \mathbb R$, where ${\rm T}\mathbb R^n = \mathbb R^{2n}$ is the tangent bundle of the configuration space $\mathbb R^n$. Such a function determines the **Euler-Lagrange** equations:
$$ \frac{d}{dt}\left[ \frac{\partial L}{\partial v^i}\bigl( \dot\gamma(t), \gamma(t)\bigr) \right] - \frac{\partial L}{\partial q^i} \bigl( \dot\gamma(t), \gamma(t)\bigr) = 0$$
Here $(v^i,q^i)$ for $i=1,\dots,n$ are the standard coordinates on ${\rm T}\mathbb R^n$, $\gamma: [0,T] \to \mathbb R^n$ is a smooth function, and $\dot\gamma^i(t) = \frac{d\gamma^i}{dt}$. Suppose that the matrix $\frac{\partial^2 L}{\partial v^i\partial v^j}(v,q)$ is invertible for any $(v,q) \in {\rm T}\mathbb R^n$. Then the Euler-Lagrange equations are a nondegenerate second-order differential equation on $\mathbb R^n$. I am interested in the **boundary-value problem** for $L$. Namely, fix $T > 0$ and $q_1,q_2 \in \mathbb R^n$; the BVP asks to find the set $C(q_1,q_2,T)$ of all paths $\gamma: [0,T] \to \mathbb R^n$ with $\gamma(0) = q_1$ and $\gamma(t) = q_2$. Generically, this is a discrete set.

### My question

Suppose that if instead of the Euler-Lagrange equations above, I pick some small parameter $\epsilon$ and consider the differential equation $$ \frac{d}{dt}\left[ \frac{\partial L}{\partial v^i}\bigl( \dot\gamma(t), \gamma(t)\bigr) \right] - \frac{\partial L}{\partial q^i} \bigl( \dot\gamma(t), \gamma(t)\bigr) = \epsilon\gamma^{(4)}(t)^i $$ where $\gamma^{(4)}(t)^i$ is the $i$th component of the fourth derivative of $\gamma$ with respect to $t$ (the "jounce", a word I just learned from Wikipedia). For $\epsilon \neq 0$, the EL equations are a nondegenerate fourth-order differential equation, and so generically solutions to the boundary-value-problem above form a two-dimensional family. To restrict to a discrete set, we should fix more boundary values. Pick $(v_1,q_1), (v_2,q_2) \in {\rm T}\mathbb R^n$ and $T > 0$, and define $C_\epsilon(v_1,q_1,v_2,q_2,T)$ to be the set of solutions $\gamma$ to the $\epsilon$-dependent EL equations with $(\dot\gamma(0),\gamma(0)) = (v_1,_1)$ and $(\dot\gamma(T),\gamma(T)) = (v_2,q_2)$.

My question is: as $\epsilon \to 0$, in what sense do we have $C_\epsilon(v_1,q_1,v_2,q_2,T) \to C(q_1,q_2,T)$?