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Lagrangian mechanics problem

  1. Apr 21, 2008 #1
    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].
     
  2. jcsd
  3. Apr 21, 2008 #2
    I just want to mention that u may think of the constraints:

    [tex]C\sum_{i=m+1} (q_i)^2[/tex]

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

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

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

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

    best regards
    rayo
     
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