This question is closely related to another one I asked recently, and may be thought of as a warm-up to that one.

Consider $\mathbb R^n$ with its usual metric, and pick a one-form $b$ and a function $c$. Let $m$ be a positive constant, and consider the second-order differential equation for a function $q(t)$ $$ m\ddot q = db \cdot \dot q + dc $$ where I have used the metric to identify vectors and covectors, $dc$ is the differential of $c$, and $db$ is the exterior derivative of $b$ (it is contracted with $\dot q$ to yield a covector). In coordinates, and using Einstein's summation convention: $$ m\ddot q^i = \left(\partial\_i b\_j - \partial\_j b\_i\right)\dot q^j + \partial\_i c $$

I am interested in the limit when $m\to 0$. For example, when $m=0$ and $b=0$ (or anyway when $b$ is closed), then the differential equation forces the path $q(t)$ to stay within the set of critical points of $c$ (this set is generically discrete, so that the only solutions are constant). At another (more generic) extreme, $db$ might be nondegenerate, and hence a symplectic form on $\mathbb R^n$. Then the equation $0 = db \cdot \dot q + dc$ is a nondegenerate first-order differential equation, exactly equivalent to Hamilton's equations for the symplectic manifold $(\mathbb R^n,db)$ with Hamiltonian $-c$. There is some gradation when $db$ is nonzero but has nontrivial kernel (as for example must happen if $n$ is odd).

So I basically get what happens when $m=0$. But can we understand the limit $m\to 0$? For example, if $m\neq 0$, then any initial value $(\dot q(0),q(0))$ determines a solution; for fixed initial values, how does this solution vary as $m\to 0$? Alternately, we can try to solve the boundary value problem, in which we prescribe $q(0)$ and $q(1)$. Then what happens to the solutions as $m$ shrinks? Since when $m=0$ we cannot find solutions with arbitrary initial velocity, it is unlikely that anything is particularly well-behaved in the limit, but not impossible.

Very specifically, I would like to know about the asymptotics of the solutions to the boundary and initial-value problems — what do solutions look like when $m$ is a formal variable? But more generally I'm happy with some statements about the regularity in the $m\to 0$ limit.

This question is closely related to another one I asked recently, and may be thought of as a warm-up to that one.

Consider $\mathbb R^n$ with its usual metric, and pick a one-form $b$ and a function $c$. Let $m$ be a positive constant, and consider the second-order differential equation for a function $q(t)$ $$ m\ddot q = db \cdot \dot q + dc $$ where I have used the metric to identify vectors and covectors, $dc$ is the differential of $c$, and $db$ is the exterior derivative of $b$ (it is contracted with $\dot q$ to yield a covector). In coordinates, and using Einstein's summation convention: $$ m\ddot q^i = \left(\partial\_i b\_j - \partial\_j b\_i\right)\dot q^j + \partial\_i c $$

I am interested in the limit when $m\to 0$. For example, when $m=0$ and $b=0$ (or anyway when $b$ is closed), then the differential equation forces the path $q(t)$ to stay within the set of critical points of $c$ (this set is generically discrete, so that the only solutions are constant). At another (more generic) extreme, $db$ might be nondegenerate, and hence a symplectic form on $\mathbb R^n$. Then the equation $0 = db \cdot \dot q + dc$ is a nondegenerate first-order differential equation, exactly equivalent to Hamilton's equations for the symplectic manifold $(\mathbb R^n,db)$ with Hamiltonian $-c$. There is some gradation when $db$ is nonzero but has nontrivial kernel (as for example must happen if $n$ is odd).

So I basically get what happens when $m=0$. But can we understand the limit $m\to 0$? For example, if $m\neq 0$, then any initial value $(\dot q(0),q(0))$ determines a solution; for fixed initial values, how does this solution vary as $m\to 0$? Alternately, we can try to solve the boundary value problem, in which we prescribe $q(0)$ and $q(1)$. Then what happens to the solutions as $m$ shrinks? Since when $m=0$ we cannot find solutions with arbitrary initial velocity, it is unlikely that anything is particularly well-behaved in the limit, but not impossible.

Very specifically, I would like to know about the asymptotics of the solutions to the boundary and initial-value problems — what do solutions look like when $m$ is a formal variable? But more generally I'm happy with some statements about the regularity in the $m\to 0$ limit.