- #1

- 21

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there is a manifold [tex]N[/tex] of dimension [tex]n[/tex]. inside [tex]N[/tex] there is another submanifold [tex]M[/tex] of dimension [tex]m\leq n[/tex].

let [tex]\{q_i\}[/tex] be a coordinate system over [tex]N[/tex] such that [tex]q_i = 0[/tex] for

[tex]i = m+1,...,n[/tex] if the point given by [tex]\vec{q}[/tex] is in [tex]M[/tex].

let [tex]L(\vec{q})[/tex] be a lagrangian over [tex]N[/tex].

and let:

[tex]

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

[/tex]

another new lagrangian over [tex]N[/tex] where [tex]C>0[/tex].

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

[tex]

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

[/tex]

that converges point wise in time. where [tex]\vec{q}_C(t)[/tex] its the trayectory obtained from

[tex]L_C(\vec{q})[/tex] for some (any) initial conditions over [tex]M[/tex].