- #1
rayohauno
- 21
- 0
the problem formulation is the next:
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].
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].