How Does Increasing C Affect Trajectories in Lagrangian Mechanics?

[rayohauno]

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:

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

another new lagrangian over N where C&gt;0.

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

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

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.

 

[rayohauno]

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