# On Dynamics of Jellet's Egg. Asymptotic Solutions Revisited

*2020, Volume 25, Number 1, pp. 40-58*

Author(s):

**Rauch-Wojciechowski S., Przybylska M.**

We study here the asymptotic condition $\dot E=-\mu g_n {\boldsymbol v}_A^2=0$ for an eccentric rolling and sliding ellipsoid with axes of
principal moments of inertia directed along geometric axes of the ellipsoid, a rigid body which we call here Jellett's egg (JE). It is shown by using dynamic equations expressed in terms of Euler angles that the asymptotic condition is satisfied by stationary solutions.
There are 4 types of stationary solutions: tumbling, spinning, inclined rolling and
rotating on the side solutions.
In the generic situation of tumbling solutions concise explicit formulas for stationary angular velocities
$\dot\varphi_{\mathrm{JE}}(\cos\theta)$, $\omega_{3\mathrm{JE}}(\cos\theta)$ as functions of JE parameters
$\widetilde{\alpha},\alpha,\gamma$ are given. We distinguish the case $1-\widetilde{\alpha}<\alpha^2<1+\widetilde{\alpha}$, $1-\widetilde{\alpha}<\alpha^2\gamma<1+\widetilde{\alpha}$
when velocities $\dot\varphi_{\mathrm{JE}}$, $\omega_{3\mathrm{JE}}$ are defined for the whole range of inclination angles $\theta\in(0,\pi)$. Numerical simulations illustrate
how, for a JE launched almost vertically with $\theta(0)=\tfrac{1}{100},\tfrac{1}{10}$, the inversion of the JE depends
on relations between parameters.

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