We can
analyze the motion of a spinning top using the Lagrange equations for the Euler angles.
Let us assume that the top has its lowest point (tip) fixed on a surface.
We will
use the fixed point as the origin.
The rotation about the origin will be described by the Euler angles so that
all the kinetic energy
is contained in the rotation.
For a symmetric top, we can immediately write things in terms of the
rotation about principal axes of inertia.
(Remember we have three principal moments of inertia but they don't make up a vector.)
We have already written
in terms of the Euler angles.
Recalling that
is the angle between the Inertial
axis and the
axis in the rotating frame,
the
potential energy can be written,
where
is the height of the center of mass of the top above the fixed tip.
Note that we are assuming that the
symmetry axis of the top is the
axis
so that
.
We can now write the kinetic energy in terms of the Euler angles.
Since the Lagrangian does not depend on
or
(cyclic), so the momenta are conserved.
This is the angular momentum about the
axis.
This is the angular momentum about the
axis.
This is reasonable since one can see that the torque is along the line of nodes.
The actual values of
and
are set by
initial conditions in the problem.
So
and
are constants of the motion and we can solve the equations for
and
.
There is a
third Lagrange equation but it will be easier to understand the motion of the top by using the
total energy equation, along with the two conserved momenta.
This is very much
like a central force problem with the mass oscillating back and forth in the potential.
goes to zero at the limits.
The motion will be limited between the some angles
and
at which
.
This oscillation of
as the angular momentum precesses is called
nutation.
Jim Branson
2012-10-21