### 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](http://en.wikipedia.org/wiki/Jounce)). 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 $C\_\epsilon(v\_1,q\_1,v\_2,q\_2,T) \to C(q\_1,q\_2,T)$?



What happens to the solutions of a fourth-order boundary-value problem as you turn off the fourth-order coefficient?
I believe that the method of solution to your problem is called the method of "dominant balance", and in this case, "singular dominant balance." If you do a web search for that, you should be able to find the information you need.


This method will you give a perturbative solution to as high a degree as you have the propensity to calculate. You can analyze this solution to answer various questions that you implied in your original question, such as what the decay behavior of the solution is, which continuity and smoothness properties it has, etc...


If you want to study the solutions of a large class of coefficient functions, not just a specific set, you can leave arbitrary constants in a solution "ansatz" and then develop a parameterized family of solutions. Note that the algebraic expressions involved in finding the simple-looking solutions grow exponentially in the number terms which end up simplifying in the end. Computer algebra is needed to find the simplified form of these solutions, lest you go mad and kill many trees.


You may also want to search for "catastrophe theory", which catalogs the types of bifurcations that happen in systems such as you have described. This is a one-dimensional bifurcation problem, which are well-studied. 


