Unstable Top
المؤلف:
Sidney B. Cahn Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 24
2-8-2016
1804
Unstable Top
A top of mass M is spinning about a fixed point under gravity, and its axis is vertical 
but the angular velocity around its axis ω3 is insufficient for stability in that position. The Lagrangian for a top is
Where θ, φ, ψ are the usual Euler angles, I1 and I3 are the moments of
Figure 1.1
inertia about their respective axes, N is the line of nodes, and l is the distance from the point of the top O to the center of mass C (see Figure 1.1).
a) Derive all the first integrals of the motion and evaluate them in terms of the given initial conditions.
b) Show that the head will descend to an angle θ given by
c) Show that the time dependence of this θ is given by the solution of
You do not need to solve for θ(t).
SOLUTION
a) There are two integrals of motion in the generalized momenta pφ, pѱ
(1)
(2)
where we used the fact that 
is the angular velocity of the top around its axis. Applying the initial conditions
to (1) and (2), we obtain
Another integral of motion is, of course, the energy; again using the initial conditions, we have
(3)
b) From (3) and using the condition that the head will descend to a maximum angle θ where 
we have
(4)
On the other hand, from (1),
(5)
By equating 
in (4) and (5) and using the half angle formulas
we get
c) Again using (3) and (5), we have
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