Rolling of a rigid body on a plane and sphere. Hierarchy of dynamics (Q1860548)

The authors study the existence of invariant measure, additional first integrals, and Poisson structure in the problem of a rigid body rolling without sliding on a plane or sphere. The problem of a rigid body motion on a plane was studied by S. A. Chaplygin, P. Appell, and D. Korteweg. They showed that the equations of motion are reduced to a second-order linear differential equation in the case when the surface of dynamically symmetric body is a surface of revolution. It turns out that after some change of time defined by reducing multiplier, the reduced system is a Hamiltonian system. The authors consider different cases when the integrals and invariant measure can be presented as finite algebraic expressions. They also consider a generalized problem of rolling of dynamically nonsymmetric Chaplygin ball.