In his Mathematical Methods of Classical Mechanics, V.I. Arnold states the following theorem without proof in pages 75-76:
Let $\gamma$ be a smooth plane curve, and let $q_1, q_2$ be local coordinates where $q_1$ parameterises the curve and $q_2$ is the coordinate in the orthogonal direction to the curve. Consider the system with potential energy $$ U_N = Nq_2^2 + U_0(q_1, q_2). $$ and let $\varphi(t, N)$ be the evolution of $q_1$ under this potential when the system has initial conditions $$q_1(0) = q_1^0,\quad\dot{q}_1(0) = \dot{q}_1^0,\quad q_2(0) = 0,\quad\dot{q}_2(0) = 0 .$$ Then we have that the limit $\psi(t) = \lim_{N \to \infty} \varphi(t, N)$ exists, and $q_1 = \psi(t)$ satisfies Lagrange's equation $$ \frac{d}{dt} \frac{\partial L_{*}}{\partial \dot{q}_1} = \frac{\partial L_{*}}{\partial q_1} $$ with the Lagrangian $$L_{*}(q_1, \dot{q}_1) = T|_{q_2 = \dot{q}_2 = 0} - U_0|_{q_2 = 0}.$$
Arnold writes that the proof is based on the fact that $q_2 \leq c N^{-1/2}$ for some constant $c$ (depending only on the initial conditions), which is not difficult to see. I assume that $T$ is the standard kinetic energy $T = \frac{m}{2}\langle\dot{x}, \dot{x}\rangle$ (here $x$ is position). In the theorem statement, Arnold only says it is the "kinetic energy of motion along $\gamma$", but the definition in page 77 seems to support my interpretation.
My reading of the $q$ coordinates is as follows: we have a (local) parameterisation $\gamma = \gamma(q_1)$ of $\gamma$, and we define local curvilinear coordinates $(q_1, q_2)$ by $x(q) = \gamma(q_1) + q_2 n(q_1)$, where $n(q_1)$ is a unit vector orthogonal to the tangent at $q_1$, $\gamma'(q_1)$. Thus, assuming that $q_1$ is arc length, $$ \dot{x} = (1 - k(q_1)q_2)\gamma'(q_1)\dot{q}_1 + n(q_1)\dot{q}_2, $$ where $k$ is the signed curvature of $\gamma$.
Hence, with $T = \frac{m}{2}\langle\dot{x}, \dot{x}\rangle$, in terms of the $q$-coordinates we get $$T(q, \dot{q}) = \frac{m}{2}(1 - k(q_1)q_2)^2\dot{q}_1^2 + \frac{m}{2}\dot{q}_2^2. $$
The equation of motion for $q_1$ becomes $$ m a(q)\ddot{q}_1 + \frac{m}{2}\frac{\partial a}{\partial q_1}\dot{q}^2_1 + m\frac{\partial a}{\partial q_2}\dot{q}_1\dot{q}_2 = -\frac{\partial U_0(q_1, q_2)}{\partial q_1}, $$ where $a(q) = 1 - k(q_1)q_2$. Note that $$ \frac{\partial a}{\partial q_2} = -2(1 - k(q_1)q_2)k(q_1), $$ so the $\dot{q}_2$ term does not go to zero in general.
Now $\dot{q}_2$ does not seem to converge to zero: in fact, the potential $Nq_2^2$ will induce oscillations in $q_2$ of frequency of order $N$, which means that $\dot{q}_2$ is only bounded. I thus have a hard time seeing how to prove the convergence claimed by Arnold.
We do have that $a(q_1, q_2 = 0) = 1$, so that the Lagrangian converges to $L_{*}$ along the trajectories with the given initial conditions. Can one use this to prove Arnold's theorem rigorously? We can make as strong assumptions about $U_0$ as we like.