A qualitative analysis of the motion of a heavy solid of revolution on an absolutely rough plane

An inhomogeneous Chaplygin sleigh

“…Аналогичный результат содержится в работе [21], где показано, что для тела вра-щения, катящегося по абсолютно шероховатой плоскости, точка контакта в отсутствие резонансов совершает сложное движение: она периодически движется по некоторой кривой, которая вращает-ся, как твердое тело, с постоянной угловой скоростью вокруг неподвижной точки; при резонансах оказывается возможным вековой уход.…”

Untitled

“…Thus, for each of the angles the dependence on time is defined as an integral of a periodic function with the frequency ω θ , hence it can be presented in the form (see, for example, [17], [24])…”

“…Following papers [20], [24] we present the equation for the velocity of the point of contact in the forṁ…”

“…Thus, if at ω ψ + nω θ = 0 we use the frame of references rotating about the point Z 0 with the angular velocity ω ψ , then the point of contact traces some closed curve (see [24], [20]). Various types of such closed curves and trajectories corresponding to them in the fixed space are presented in figure 8.…”

Phase Space of Rolling Solutions of the Tippe Top

“…A more general discussion of the Smale diagram for rolling solutions of an ellipsoid of revolution has been presented on [14]. A qualitative analysis of motion of a solid of revolution on an absolutely rough plane has been also performed in [11] by starting from canonical equations with nonholonomicity term and by referring to properties of the monodromy matrix. These results, when specialised to the case of sphere, lead to the same picture as presented here.…”