Lagrangian mechanics problem

the problem formulation is the next:

there is a manifold $$N$$ of dimension $$n$$. inside $$N$$ there is another submanifold $$M$$ of dimension $$m\leq n$$.

let $$\{q_i\}$$ be a coordinate system over $$N$$ such that $$q_i = 0$$ for
$$i = m+1,...,n$$ if the point given by $$\vec{q}$$ is in $$M$$.

let $$L(\vec{q})$$ be a lagrangian over $$N$$.

and let:

$$L_C(\vec{q}) = L(\vec{q}) + C\sum_{i=m+1}^n (q_i)^2$$

another new lagrangian over $$N$$ where $$C>0$$.

then the problem is to show that there exist a limiting trayectory:

$$\vec{q}_{lim}(t) = lim_{C \rightarrow \infty} \,\, \vec{q}_C(t)$$

that converges point wise in time. where $$\vec{q}_C(t)$$ its the trayectory obtained from
$$L_C(\vec{q})$$ for some (any) initial conditions over $$M$$.

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I just want to mention that u may think of the constraints:

$$C\sum_{i=m+1} (q_i)^2$$

as if at any point in $$N$$ there exist some kind of springs that tend to move the particle point to
a some position in $$M$$.

by definition of the properties of the coordinate system chosen, at each point in $$N$$ the springs
forces acts on directions normal to the manifold $$M$$ (if the particle point its close enough to $$M$$).

so, enlarging $$C$$ just makes those springs more stronger, (hope) forcing to the particle live in $$M$$ in the limiting case $$C \rightarrow \infty$$.

I don´t know if the problem is positively probable, or if there exist a counter example.

best regards
rayo