This is a pendulum in 3D.
Let the angle between the pendulum and the vertical line is and the angular velocity
And the angle of the pendulum (projected to x-y plane) with the x-axis is , and it's angular velocity
The [url=http://en.wikipedia.org/wiki/Lagrangian_mechanics]lagrange equation[/url] for the system is
The equation of the motion is
Angular momentum is conserved. ...... from
And the following is the simulation of such a system:
When the checkbox ([b]circular loop[/b]) is checked, . and It is a circular motion.
The vertical component tangential of the string balanced with the mass m, and the horizontal component tangential provide the centripetal force for circular motion.
You can uncheck it and change the period ,
and you will find out the z-coordinate of the pendulum will change with time when
You can also drag the blue dot to change the length of the pendulum.