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user9072
user9072

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 ![$C(q_1,q_2,T)$](http://latex.mathoverflow.net/png?%24C%28q%5F1%2Cq%5F2%2CT%29%24) 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 $C_\epsilon(v_1,q_1,v_2,q_2,T) \to C(q_1,q_2,T)$?

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 ![$C(q_1,q_2,T)$](http://latex.mathoverflow.net/png?%24C%28q%5F1%2Cq%5F2%2CT%29%24) 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 $C_\epsilon(v_1,q_1,v_2,q_2,T) \to C(q_1,q_2,T)$?

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 $C_\epsilon(v_1,q_1,v_2,q_2,T) \to C(q_1,q_2,T)$?

Eliminating use of latex.mathoverflow.net, per http://meta.mathoverflow.net/a/385
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edit approved Jul 7, 2013 at 23:55
Jeremy
edit approved Jul 7, 2013 at 23:55
Jeremy
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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 ![$q\sb 1,q\sb 2 \in \mathbb R^n$](http://latex.mathoverflow.net/png?%24q%5F1%2Cq%5F2%20%5Cin%20%5Cmathbb%20R%5En%24)$q_1,q_2 \in \mathbb R^n$; the BVP asks to find the set ![$C(q\sb 1,q\sb 2,T)$$C(q_1,q_2,T)$](http://latex.mathoverflow.net/png?%24C%28q%5F1%2Cq%5F2%2CT%29%24) of all paths $\gamma: [0,T] \to \mathbb R^n$ with ![$\gamma(0) = q\sb 1$](http://latex.mathoverflow.net/png?%24%5Cgamma%280%29%20%3D%20q%5F1%24)$\gamma(0) = q_1$ and ![$\gamma(t) = q\sb 2$](http://latex.mathoverflow.net/png?%24%5Cgamma%28t%29%20%3D%20q%5F2%24)$\gamma(t) = q_2$. 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 ![$(v\sb 1,q\sb 1), (v\sb 2,q\sb 2) \in {\rm T}\mathbb R^n$](http://latex.mathoverflow.net/png?%24%28v%5F1%2Cq%5F1%29%2C%20%28v%5F2%2Cq%5F2%29%20%5Cin%20%7B%5Crm%20T%7D%5Cmathbb%20R%5En%24)$(v_1,q_1), (v_2,q_2) \in {\rm T}\mathbb R^n$ and $T > 0$, and define ![$C\sb \epsilon(v\sb 1,q\sb 1,v\sb 2,q\sb 2,T)$](http://latex.mathoverflow.net/png?%24C%5F%5Cepsilon%28v%5F1%2Cq%5F1%2Cv%5F2%2Cq%5F2%2CT%29%24)$C_\epsilon(v_1,q_1,v_2,q_2,T)$ to be the set of solutions $\gamma$ to the $\epsilon$-dependent EL equations with ![(\dot\gamma(0),\gamma(0)) = (v\sb 1,q\sb 1)](http://latex.mathoverflow.net/png?%28%5Cdot%5Cgamma%280%29%2C%5Cgamma%280%29%29%20%3D%20%28v%5F1%2Cq%5F1%29)$(\dot\gamma(0),\gamma(0)) = (v_1,_1)$ and ![(\dot\gamma(T),\gamma(T)) = (v\sb 2,q\sb 2)](http://latex.mathoverflow.net/png?%28%5Cdot%5Cgamma%28T%29%2C%5Cgamma%28T%29%29%20%3D%20%28v%5F2%2Cq%5F2%29)$(\dot\gamma(T),\gamma(T)) = (v_2,q_2)$.

My question is: as $\epsilon \to 0$, in what sense do we have C\sb \epsilon(v\sb 1,q\sb 1,v\sb 2,q\sb 2,T) \to C(q\sb 1,q\sb 2,T) http://latex.mathoverflow.net/png?C%5F%5Cepsilon%28v%5F1%2Cq%5F1%2Cv%5F2%2Cq%5F2%2CT%29%20%5Cto%20C%28q%5F1%2Cq%5F2%2CT%29$C_\epsilon(v_1,q_1,v_2,q_2,T) \to C(q_1,q_2,T)$?

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 ![$q\sb 1,q\sb 2 \in \mathbb R^n$](http://latex.mathoverflow.net/png?%24q%5F1%2Cq%5F2%20%5Cin%20%5Cmathbb%20R%5En%24); the BVP asks to find the set ![$C(q\sb 1,q\sb 2,T)$](http://latex.mathoverflow.net/png?%24C%28q%5F1%2Cq%5F2%2CT%29%24) of all paths $\gamma: [0,T] \to \mathbb R^n$ with ![$\gamma(0) = q\sb 1$](http://latex.mathoverflow.net/png?%24%5Cgamma%280%29%20%3D%20q%5F1%24) and ![$\gamma(t) = q\sb 2$](http://latex.mathoverflow.net/png?%24%5Cgamma%28t%29%20%3D%20q%5F2%24). 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 ![$(v\sb 1,q\sb 1), (v\sb 2,q\sb 2) \in {\rm T}\mathbb R^n$](http://latex.mathoverflow.net/png?%24%28v%5F1%2Cq%5F1%29%2C%20%28v%5F2%2Cq%5F2%29%20%5Cin%20%7B%5Crm%20T%7D%5Cmathbb%20R%5En%24) and $T > 0$, and define ![$C\sb \epsilon(v\sb 1,q\sb 1,v\sb 2,q\sb 2,T)$](http://latex.mathoverflow.net/png?%24C%5F%5Cepsilon%28v%5F1%2Cq%5F1%2Cv%5F2%2Cq%5F2%2CT%29%24) to be the set of solutions $\gamma$ to the $\epsilon$-dependent EL equations with ![(\dot\gamma(0),\gamma(0)) = (v\sb 1,q\sb 1)](http://latex.mathoverflow.net/png?%28%5Cdot%5Cgamma%280%29%2C%5Cgamma%280%29%29%20%3D%20%28v%5F1%2Cq%5F1%29) and ![(\dot\gamma(T),\gamma(T)) = (v\sb 2,q\sb 2)](http://latex.mathoverflow.net/png?%28%5Cdot%5Cgamma%28T%29%2C%5Cgamma%28T%29%29%20%3D%20%28v%5F2%2Cq%5F2%29).

My question is: as $\epsilon \to 0$, in what sense do we have C\sb \epsilon(v\sb 1,q\sb 1,v\sb 2,q\sb 2,T) \to C(q\sb 1,q\sb 2,T) http://latex.mathoverflow.net/png?C%5F%5Cepsilon%28v%5F1%2Cq%5F1%2Cv%5F2%2Cq%5F2%2CT%29%20%5Cto%20C%28q%5F1%2Cq%5F2%2CT%29?

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 ![$C(q_1,q_2,T)$](http://latex.mathoverflow.net/png?%24C%28q%5F1%2Cq%5F2%2CT%29%24) 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 $C_\epsilon(v_1,q_1,v_2,q_2,T) \to C(q_1,q_2,T)$?

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Theo Johnson-Freyd
Theo Johnson-Freyd
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What happens to the solutions of a fourth-order boundary-value problem as you turn off the fourth-order coefficient?

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 ![$q\sb 1,q\sb 2 \in \mathbb R^n$](http://latex.mathoverflow.net/png?%24q%5F1%2Cq%5F2%20%5Cin%20%5Cmathbb%20R%5En%24); the BVP asks to find the set ![$C(q\sb 1,q\sb 2,T)$](http://latex.mathoverflow.net/png?%24C%28q%5F1%2Cq%5F2%2CT%29%24) of all paths $\gamma: [0,T] \to \mathbb R^n$ with ![$\gamma(0) = q\sb 1$](http://latex.mathoverflow.net/png?%24%5Cgamma%280%29%20%3D%20q%5F1%24) and ![$\gamma(t) = q\sb 2$](http://latex.mathoverflow.net/png?%24%5Cgamma%28t%29%20%3D%20q%5F2%24). 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 ![$(v\sb 1,q\sb 1), (v\sb 2,q\sb 2) \in {\rm T}\mathbb R^n$](http://latex.mathoverflow.net/png?%24%28v%5F1%2Cq%5F1%29%2C%20%28v%5F2%2Cq%5F2%29%20%5Cin%20%7B%5Crm%20T%7D%5Cmathbb%20R%5En%24) and $T > 0$, and define ![$C\sb \epsilon(v\sb 1,q\sb 1,v\sb 2,q\sb 2,T)$](http://latex.mathoverflow.net/png?%24C%5F%5Cepsilon%28v%5F1%2Cq%5F1%2Cv%5F2%2Cq%5F2%2CT%29%24) to be the set of solutions $\gamma$ to the $\epsilon$-dependent EL equations with ![(\dot\gamma(0),\gamma(0)) = (v\sb 1,q\sb 1)](http://latex.mathoverflow.net/png?%28%5Cdot%5Cgamma%280%29%2C%5Cgamma%280%29%29%20%3D%20%28v%5F1%2Cq%5F1%29) and ![(\dot\gamma(T),\gamma(T)) = (v\sb 2,q\sb 2)](http://latex.mathoverflow.net/png?%28%5Cdot%5Cgamma%28T%29%2C%5Cgamma%28T%29%29%20%3D%20%28v%5F2%2Cq%5F2%29).

My question is: as $\epsilon \to 0$, in what sense do we have C\sb \epsilon(v\sb 1,q\sb 1,v\sb 2,q\sb 2,T) \to C(q\sb 1,q\sb 2,T) http://latex.mathoverflow.net/png?C%5F%5Cepsilon%28v%5F1%2Cq%5F1%2Cv%5F2%2Cq%5F2%2CT%29%20%5Cto%20C%28q%5F1%2Cq%5F2%2CT%29?