Question 13.12: Use the Euler equations to show that for an asymmetric top t...
Use the Euler equations to show that for an asymmetric top the rotations about the axes of the largest and smallest moment of inertia are stable; however, the rotation about the axis of the intermediate moment of inertia is unstable.
We start from the Euler equations for the free top:
\dot{ω}_{1} = \frac{Θ_{2} −Θ_{3}}{Θ_{1}} ω_{2}ω_{3}, (13.119)
\dot{ω}_{2} = \frac{Θ_{3} −Θ_{1}}{Θ_{2}} ω_{1}ω_{3}, (13.120)
\dot{ω}_{3} = \frac{Θ_{1} −Θ_{2}}{Θ_{3}} ω_{1}ω_{2}, (13.121)
Let the top rotate about the body-fixed z-axis, i.e., ω_{3} = ω_{0} = constant and ω_{1} = ω_{2} = 0. To investigate the stability of the rotation about this principal axis, we tilt the rotation axis by a small amount, so that new components δω_{1}, δω_{2} and an additional δω_{3} arise. For δ\dot{ω}_{3}, we have from the Euler equation
δ\dot{ω}_{3}= \frac{Θ_{1} −Θ_{2}}{Θ_{3}}δω_{1}δω_{2}\simeq 0. (13.122)
Neglecting quadratic small terms, we can set ω_{3} = ω_{0}. From the other two Euler equations, we then obtain
δ\dot{ω}_{1} + \frac{Θ_{3} −Θ_{2}}{Θ_{1}} δω_{2}ω_{0} = 0, (13.123)
δ\dot{ω}_{2} + \frac{Θ_{1} −Θ_{3}}{Θ_{2}} δω_{1}ω_{0} = 0, (13.124)
To solve this coupled system, we use the ansatz
δω_{1} = Ae^{λt} ,
δω_{2} = Be^{λt} . (13.125)
This leads to a linear set of equations in A and B, where the determinant must vanish for nontrivial solutions:
\begin{vmatrix} λ & \frac{Θ_{3} −Θ_{2}}{Θ_{1}}ω_{0} \\ \frac{Θ_{1} −Θ_{3}}{Θ_{2}}ω_{0} & λ \end{vmatrix}= 0. (13.126)
From this, we find the characteristic equation
λ^{2} = ω^{2}_{0} \frac{(Θ_{3} −Θ_{2})(Θ_{1} −Θ_{3})}{Θ_{1}Θ_{2}}. (13.127)
For the rotation about the axis of the smallest moment of inertia Θ_{3} <Θ_{1},Θ_{2}, and for the rotation about the axis of the largest moment of inertia Θ_{3} >Θ_{1},Θ_{2}, equation (13.127) leads to a purely imaginary λ:
λ^{2} < 0, (13.128)
and therefore to vibration solutions for δω_{1} and δω_{2}. The rotation about the axis of the largest and smallest moment of inertia, respectively, is therefore stable.
The rotation about the axis of the intermediate moment of inertia
Θ_{1} >Θ_{3} >Θ_{2} or Θ_{2} >Θ_{3} >Θ_{1} (13.129)
leads to a real λ and thus to a time evolution of δω_{1} and δω_{2} according to
δω_{1/2} = C_{1/2} cosh λt +D_{1/2} sinh λt. (13.130)
The rotation axis turns away exponentially from the initial position. The rotation about the axis of the intermediate moment of inertia is not stable!