How to analyze this constraint question

Let $\gamma$ be a smooth curve in the plane, and introduce curvilinear coordinates $q_1,q_2$ on a neighborhood of $\gamma$; $q_1$ is the direction of $\gamma$ and $q_2$ is distance from the curve. Consider the system with potential energy $$U_N = Nq_2^2+U_0(q_1,q_2)$$ depending on the parameter $N$. And the initial conditions are $$q_1(0) = q_1^0, \qquad \dot{q}_1(0)=\dot{q}_1^0,\qquad q_2(0) = 0, \qquad\dot{q}_2(0)=0$$ Denote $q_2 = g(t,N)$ the evolution of the coordinate $q_2$ under a motion with these initial conditions in the field $U_N$, then the question is to prove $$\lim_{N\rightarrow \infty} g(t,N) = 0$$ I have tried as following: Assume the motion is with the whole energy $E = T + U_N$, then $$q_2^2 = \frac{E - T - U_0(q_1,q_2)}{N}$$ If $U_0(q_1,q_2) \geq U_{min}$, then we know $$q_2^2 \leq \frac{E - T - U_{min}}{N}$$ which tends to zero as $N\rightarrow \infty$ for any $t$. But what if there is no lower bound for $U_0(q_1,q_2)$?

• Since this is a homework-style question, what have you tried thus far? – joshphysics Nov 19 '13 at 2:55