Spinning tops need to be visualised in reference to 2 frames - the space frame ( *x* , *y* , *z* ) that is effectively the
'real life' frame that we have been using up until now when describing the position of the pendulums; and the
body frame ( *x'* , *y'* , *z'* ) that is basically another set of ( *x* , *y* , *z* ) axes taken with the *z'*-axis pointing up through
the middle of the spinning top. This body frame obviously changes its orientation with respect to the space
frame as the spinning top moves.

There are 3 degrees of freedom needed to describe the position of the spinning top - ,
which is the angle between the *z*-axis of the space frame and the *z'*-axis of the body frame; ,
which is the rotation angle of the *z'*-axis of the body frame around the *z*-axis of the space frame; and ,
which is the rotation angle of the spinning top around its own *z'*-axis (that of the body frame).

A diagram of the symmetric spinning top is shown below:

Lagrange's equations are needed again to derive the equations of motion used to model the
spinning top. The equations of motion for the symmetric top are shown below:

(Click __Here__ to see the proof)

The applet below simulates the motion of the symmetric spinning top:

It is possible to adjust the initial values for , ,
, , and .
It is also possible to alter the values for the mass *m* of the top, and *l*, which is the length of the top along the
*z'*-axis from the origin (point of the top) to the centre of the circle at the other end of the top.

In addition to this, you can activate/deactivate gravity; adjust the framerate of the animation (if you want to watch the applet in slow motion, for example); and turn a reference
dot on/off. This reference dot highlights a fixed point on the top, and is designed to help visualise how fast the top is spinning.

© Michael Hart 2004