The spinning motion of a symmetric top.
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A symmetric top started spinning about a vertical axis. In order not to topple over it must be spinning sufficiently fast. How fast is sufficiently fast? Provide representative sketches for the effective potential for teh case of stable and non-stable motion. When the top is not spinning fast enough to remain spinning in the vertical position it starts nutating. Find the nutation angles.
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The Euler's equations are:
(1.1)
Thus:
(1.2)
The kinetic energy of the system in terms of the Euler's angles is:
(1.3)
The potential energy is:
(1.4)
Thus, the Lagrangian is:
(1.5)
The Lagrangian is independent of the generalized coordinates , thus their conjugate momentum is conserved:
(1.6)
Therefore:
And:
Also:
The Hamiltonian can be written as:
...
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