1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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:

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

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

    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 [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 donĀ“t know if the problem is positively probable, or if there exist a counter example.

    best regards
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook