Today we began some examples of how to set up and solve a mechanics problem using variational methods. That is, finding general coordinates, enumerating constraints, finding proper sets of coordinates, and using overcomplete sets of coordinates to find forces of constraint.
The first example we considered was the spherical pendulum. This is
just an ordinary pendulum, with fixed support, where the bob is
allowed to move in all directions in 3-space. As a particle in
3-space, the bob has 3 naive degrees of freedom; however, the
constraint that r=l eliminates 1 degree of freedom to give a system
with 2 degrees of freedom. We found a proper set of coordinates
, derived the Lagrangian, and wrote out the E-L
equations for 
and 
. The equation gave us
conservation of 
; the equation a slightly more complex
equation whose qualitative features we discussed. The
variable experiences two forces: the usual gravitational restoring
force for a pendulum, and a centrifugal force which wants to drive
higher as we increase the speed 
of rotation
about the z-axis. To find the tension in the rope -- which enforces
the constraint r = l -- we redid our analysis, adding back in as a
coordinate r, which describes the direction in which we expect the
tension to act. We also added a Lagrange multiplier term to enforce
the constraint. Doing E-L equations for 
and
r gave: 1) 
enforces constraint; 2) 
same as before once we plug in r=l constraint;
3) 
equation for 
. Imposing the constraint,
; solving this gives 
. We
then know that is the r-component of the tension, since
plays the role of an effective force in our
E-L equations ( 
plays the role of an effective potential
in our Lagrangian). Here 
is just 
.
--
KB