Consider a heavy spinning symmetric top with moments of inertia $I_1=I_2=I$ and $I_3$ in an inertial frame which experiences two forces:

  1. Gravitational weight force $\vec F_\text{grav}=m\vec g$ where $\vec g=-g\hat e_z$ that act through the CoM located at $\vec r=R\hat e_r$
  2. Contact force with the ground, we will assume fixed point like contact, thus it creates no torque

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🧠 Remember: we will use the following formulas derived

$$ \begin{aligned} \vec\omega&=\dot \phi[\cos(\theta)\hat e_r-\sin(\theta)\hat e_\theta]+\dot \theta\hat e_\phi \\ \dot{ \hat e}r&=\dot\theta\hat e\theta+\dot\phi \sin(\theta)\hat e_\phi \\ \dot{\hat e}\theta &= -\dot\theta \hat e_r+\dot\phi\cos(\theta)\hat e\phi \\ \dot {\hat e}\phi&=-\dot\phi[\sin(\theta)\hat e_r+\cos(\theta)\hat e\theta] \end{aligned} $$

Special cases

Consider the case where $\theta=\theta_0$ is a constant, where there can be a spinning and precessing solution with constant $\dot\psi=\Omega_S$ and $\dot \phi=\Omega_P$ with $\Omega_S=\frac{I}{I_3}a-\Omega_P\cos(\theta_0)$ and $\Omega_P$ is the solution to the following quadratic equation:

$$ \cos(\theta_0)\Omega^2_P-a\Omega_P+\frac 12 \beta=0 $$

where the solution is

$$ \Omega_P=\frac{a}{2\cos(\theta_0)}\left [ 1 \pm \left ( 1-\frac{2\beta\cos(\theta_0)}{a^2} \right )^\frac 12 \right ] $$

Cases:

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Key features