Stefan Rauch-Wojciechowski

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.

A counterintuitive unidirectional (say counterclockwise) motion of a toy rattleback takes place when it is started by tapping it at a long side or by spinning it slowly in the clockwise sense of rotation. We study the motion of a toy rattleback having an ellipsoidal-shaped bottom by using frictionless Newton equations of motion of a rigid body rolling without sliding in a plane. We simulate these equations for tapping and spinning initial conditions to see the contact trajectory, the force arm and the reaction force responsible for torque turning the rattleback in the counterclockwise sense of rotation. Long time behavior of such a rattleback is, however, quasi-periodic and a rattleback starting with small transversal oscillations turns in the clockwise direction.

We present a qualitative analysis of the dynamics of a rolling and sliding disk in a horizontal plane. It is based on using three classes of asymptotic solutions: straight-line rolling, spinning about a vertical diameter and tumbling solutions. Their linear stability analysis is given and it is complemented with computer simulations of solutions starting in the vicinity of the asymptotic solutions. The results on asymptotic solutions and their linear stability apply also to an annulus and to a hoop.

We study the relationship between numerical solutions for inverting Tippe Top and the structure of the dynamical equations. The numerical solutions confirm the oscillatory behavior of the inclination angle $\theta(t)$ for the symmetry axis of the Tippe Top, as predicted by the Main Equation for the Tippe Top. They also reveal further fine features of the dynamics of inverting solutions defining the time of inversion. These features are partially understood on the basis of the underlying dynamical equations.

Our solution to the Jacobi problem of finding separation variables for natural Hamiltonian systems $H = \frac{1}{2} p^2 + V(q)$ is explained in the first part of this review. It has a form of an effective criterion that for any given potential $V(q)$ tells whether there exist suitable separation coordinates $x(q)$ and how to find these coordinates, so that the Hamilton-Jacobi equation of the transformed Hamiltonian is separable. The main reason for existence of such criterion is the fact that for separable potentials $V(q)$ all integrals of motion depend quadratically on momenta and that all orthogonal separation coordinates stem from the generalized elliptic coordinates. This criterion is directly applicable to the problem of separating multidimensional stationary Schrodinger equation of quantum mechanics.
Second part of this work provides a summary of theory of quasipotential, cofactor pair Newton equations $\ddot q=M(q)$ admitting $n$ quadratic integrals of motion. This theory is a natural generalization of theory of separable potential systems $\ddot q=−∇V(q)$. The cofactor pair Newton equations admit a Hamilton–Poisson structure in an extended $2n + 1$ dimensional phase space and are integrable by embedding into a Liouville integrable system. Two characterizations of these systems are given: one through a Poisson pencil and another one through a set of Fundamental Equations. For a generic cofactor pair system separation variables have been found and such system have been shown to be equivalent to a Stäckel separable Hamiltonian system. The theory is illustrated by examples of driven and triangular Newton equations.

A fast rotating tippe top (TT) defies our intuition because, when it is launched on its bottom, it flips over to spin on its handle. The existing understanding of the flipping motion of TT is based on analysis of stability of asymptotic solutions for different values of TT parameters: the eccentricity of the center of mass $0 \leqslant \alpha \leqslant 1$ and the quotient of main moments of inertia $\gamma=I_1/I_3$. These results provide conditions for flipping of TT but they say little about dynamics of inversion.
I propose here a new approach to study the equations of TT and introduce a Main Equation for the tippe top. This equation enables analysis of dynamics of TT and explains how the axis of symmetry $\hat{3}$ of TT moves on the unit sphere $S^2$. This approach also makes possible to study the relationship between behavior of TT and the law of friction.

A rigorous, and possibly complete analysis of the phase space picture of the tippe top solutions for all initial conditions when the top does not jump and all relations between parameters $\alpha$ and $\gamma$, is for the first time presented here. It is based on the use the Jellett's integral of motion $\lambda$ and the analysis of the energy function. Theorems about stability and attractivity of the asymptotic manifold are proved in detail. Lyapunov stability of (periodic) asymptotic solutions with respect to arbitrary perturbations is shown.