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 an equivalent integrated form of the Tippe Top (TT) equations that leads to the Main Equation for the Tippe Top (METT), an equation describing time evolution of the inclination angle θ(t) of inverting TT. We study how the effective potential V(cos θ, D, λ) in METT deforms as TT is inverting and show that its minimum moves from a neighborhood of θ = 0 to a neighborhood of θ = π. We analyse behaviour of θ(t) and show that it oscillates and moves toward θ = π when the physical parameters of the TT satisfy 1 − α² < γ < 1 and the initial conditions are such that Jellett’s integral satisfy

. Estimates for maximal value of the oscillation period of θ(t) are given.

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 θ(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 = ½p 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 Schrödinger equation of quantum mechanics. Second part of this work provides a summary of theory of quasipotential, cofactor pair Newton equations = M(q) admitting n quadratic integrals of motion. This theory is a natural generalization of theory of separable potential systems = −∇(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 = a = 1 and the quotient of main moments of inertia ? = I1/ I3. 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 3 of TT moves on the unit sphere S2. This approach also makes possible to study the relationship between behavior of TT and the law of friction. © MAIK Nauka 2008.

Equations of motion of an axially symmetric sphere rolling and sliding on a plane are usually taken as model of the tippe top. We study these equations in the nonsliding regime both in the vector notation and in the Euler angle variables when they admit three integrals of motion that are linear and quadratic in momenta. In the Euler angle variables (θ, ϕ, ψ) these integrals give separation equations that have the same structure as the equations of the Lagrange top. It makes it possible to describe the whole space of solutions by representing them in the space of parameters (D, λ, E) being constant values of the integrals of motion.

Triangular form of Newton equations is a strong property. Together with the existence of a single quadratic with respect to velocities integral of motion, it usally implies existence of further n - 1 integrals that are also quadratic. These integrals make the triangular system separable in new type of coordinates. The separation coordinates are built of quadric surfaces that are nonorthogonal and noconfocal and can intersect along lower dimensional singular manifolds. We present here separability theory for n-dimensional triangular systems and analyze the structure of separation coordinates in two and three dimensions. © 2007 by the Massachusetts Institute of Technology.