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 p_φ, p_ѱ

(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