A type of singular limit for systems of differential equations

Question

Suppose I have a system of differential equations for the unknowns $(x_1,v_1,\ldots,x_N,v_N)$ (interpreted as the positions & velocities of $N$ labeled particles), $$\begin{cases}\dot{x}_{i,\epsilon} = v_{i,\epsilon} \\ \dot{v}_{i,\epsilon} = \displaystyle-\frac{1}{\epsilon}\sum_{1\leq j\leq N} \nabla W(x_{i,\epsilon}-x_{j,\epsilon}) \\ (x_{i,\epsilon},v_{i,\epsilon})\vert_{t=0} = (x_i^0,v_i^0), \end{cases} \qquad 1\leq i\leq N. \tag{1}$$ Above, $\epsilon>0$ is small parameter, and for simplicity, let's assume that the positions $x_{i,\epsilon}\in \mathbb{T}^d$ ($[-\frac12,\frac12]^d$ with periodic boundary conditions) and $v_{i,\epsilon}\in\mathbb{R}^d$. $W$ is some function, the exact form is not so important for formulating the question.

Multiplying both sides of the second equation by $\epsilon$ and sending $\epsilon\rightarrow 0$, I would formally expect that $(x_{i,\epsilon},v_{i,\epsilon})_{i=1}^N$ to converge to a solution $(x_i,v_i)_{i=1}^N$ of the system. $$\begin{cases}\dot{x}_i = v_i \\ \displaystyle0 = -\sum_{1\leq j\leq N}\nabla W(x_i-x_j) \\ (x_i,v_i)\vert_{t=0} = (x_i^0,v_i^0), \end{cases} \qquad 1\leq i\leq N.\tag{2}$$ Introducing the potential energy $$P = \frac12\sum_{1\leq i,j\leq N} W(x_i-x_j),$$ the second equation of (2) says that $(x_i)_{i=1}^N$ is a critical point of $P$ for all time. So the $N$ particles are just translated in space through the set of critical points of the potential energy.

I'm interested in references that have studied these singular limit problems and, in particular, the assumptions on $W$ needed to show convergence.

Answer 1

A couple of comments and some literature pointers:

First, maybe you were already aware, but such singular limit problems are studied under the name "singular perturbation theory". An introduction are the lecture notes "Geometric singular perturbation theory" by Christopher K. R. T. Jones (1995).

Further, I'd only expect the system to converge to critical points of $P$, if these points are actually local minima. Since this is a problem described by a potential, it reminds me of the article "Motion under the influence of a strong constraining force" by Floris Takens (1980) in Springer Lecture Notes in Math. vol 819. Actually with that paper in mind, solutions might not really converge to local minima of $P$, but oscillate around them instead.