This question is closely related to [another one I asked recently](https://mathoverflow.net/questions/4459/), 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.



What happens to Newtonian systems as the mass vanishes?
This is not a real answer, but there is a branch of mathematics called "semiclassical analysis" which might be related. For example, consider a degenerate version of the problem above:
$$
(-h^2 \partial\_x^2+V(x))u=0.
$$
Here $h^2=m$ and $V=dc$; we assume that $n=1$. Then the limit as $h\to 0$ is called "semiclassical limit". What should happen (if you prescribe some boundary conditions) is that the possible solutions $u$ should get "microlocalized" near the zero set $\{p=0\}$ of the semiclassical symbol
$$
p(x,\xi)=\xi^2+V(x).
$$
Here a function $u$ is "microlocalized" near a subset $K$ of the cotangent bundle if a certain norm $\|Au\|$ is small for any pseudodifferential operator $A$ with symbol $a$ supported outside of $K$.


A physical explanation of the above would be that our static Schr\"odinger equation governs the behaviour of a single quantum particle under the potential $V$ near the zero energy level; for small values of Planck constant $h$, this should correspond to the motion of a classical particle at this fixed energy level.


There are several sources to read about semiclassical analysis, including 
[lecture notes by Evans-Zworski](http://math.berkeley.edu/~zworski/semiclassical.pdf) and [a book by Dimassi and Sjostrand](http://books.google.com/books?id=2wh1Lb4F3oYC&dq=Spectral+analysis+in+the+semiclassical+limit&source=gbs_navlinks_s).


