Untitled

Abstract

In this paper we consider cases of existence of invariant measure, additional first integrals, and Poisson structure in a problem of rigid body's rolling without sliding on plane and sphere. The problem of rigid body's motion on plane was studied by S. A. Chaplygin, P. Appel, D. Korteweg. They showed that the equations of motion are reduced to a secondorder linear differential equation in the case when the surface of dynamically symmetric body is a surface of revolution. These results were partially generalized by P. Woronetz, who studied the motion of body of revolution and the motion of round disk with sharp edge on the surface of sphere. In both cases the systems are Euler -Jacobi integrable and have additional integrals and invariant measure. It turns out that after some change of time defined by reducing multiplier, the reduced system is a Hamiltonian system. Here we consider different cases when the integrals and invariant measure can be presented as finite algebraic expressions.We also consider the generalized problem of rolling of dynamically nonsymmetric Chaplygin ball. The results of studies are presented as tables that describe the hierarchy of existence of various tensor invariants: invariant measure, integrals, and Poisson structure in the considered problems.

Poisson structures for reduced non-holonomic systems

“…Thus, we have shown that the framework suggested by Borisov, Mamaev and Kilin [7,8] can be improved along the lines discussed, namely, that those reduced systems need no rescaling to become Hamiltonian with respect to a Poisson structure of rank two, and that the domain of definition of the Poisson structures introduced therein can be extended, including even the set of singular equilibria of the reduced systems. A natural question is whether a similar approach could be used in other non-holonomic systems, maybe of higher dimension.…”

“…This problem has been treated as well, e.g., in [4,7,20,33,35]. Consider a sphere of mass m and radius r with its center of mass at a distance α (0 < α < r) from its geometric center.…”

“…Section 4 is devoted to show the application of the previous results in specific examples, namely, the rolling disk, the Routh's sphere, and the ball rolling on a surface of revolution. We will use the formulation of [18], [17] and [25], respectively, of these problems, rather than that of [7,8]. However, we point out the equivalence of both treatments in the last case.…”

“…In addition, Borisov, Mamaev and Kilin [7,8] have recently found a Poisson structure for each studied case of reduced non-holonomic systems, such that the reduced system becomes Hamiltonian, with respect to such a structure, after a rescaling, the Hamiltonian function being the reduced energy. The examples treated by them are classical in the literature, consisting mainly of rolling bodies without slipping, namely a rigid body of revolution rolling on a plane, in particular the Routh's sphere (see Section 4.2), the rolling disk (to be treated in Section 4.1), the motion of a homogeneous ball on a surface of revolution (called sometimes Routh's problem, see, e.g., [42,43]), and other cases.…”

Nonlinear dynamics of the rattleback: a nonholonomic model

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