I want to change the topic slightly concerning this problem; namely as to
whether or not the problem is an intrinsically 1 dimensional or 2
dimensional problem? (I'm talking geometric dimensions, here). One group
of folks have commented that it is intrinsically 2 dimensional as evidenced
by the the changed direction of the force vectors in the tension of the
rope. Yet others, basically treat it one dimensionally, ignoring this
changed direction; and apparently to no ill effects regarding calculating
the acceleration of the objects involved and their subsequent motion.
My comment/questioned (hopefully designed to get some response) is this: It
seems to me that the situation can be regarded as intrinsically one
dimensional. After all, if I try to solve the problem using lagrangian
mechanics I need only *one* generalized coordinate and can therefore regard
the background manifold as one dimensional.
I realize that the power of the lagrangian methods, is that constraints (in
this case the finite length rope attaching the masses to each other,
non-stretchable as well; all available by mail order from the magical
introductory physics supply warehouse) are automatically built-in. Which
means that I only need one generalized coordinate for this problem rather
than the *six* (3*N) one would maximally expect for a 3 dimensional two
particle problem.
In what sense is it legitimate for me to say that the problem is therefore a
one-dimension problem. This isn't unrelated to saying the problem is
intrinsically two dimensional; rather than three, since we know the motion
is confined to a plane.