In this paper we investigate the problem of rolling of a sphere over a fixed horizontal plane; it is assumed that the sphere has a multiply connected cavity with an ideal fluid in vortex-free motion. We show that the solution of the problem can be reduced to quadrature.

An attempt is made to find a theoretical basis for some dynamic effects discovered experimentally in one problem of solid body dynamics on a plane, namely, the problem of the motion of the "celtic stone" [1-4]. The main attention is given to oscillations of a solid close to the equilibrium position or steady rotation. The motion is assumed to occur without friction and the supporting plane is fixed. Small oscillations of the body are briefly considered in the neighborhood of its steady rotation about the vertical. An approximate system of equations is obtained which describes non-linear oscillations of the body in the vicinity of its equilibrium position on a plane and a complete analysis is given. The results of the investigation agree with experimental observations [1,3] of the changes in the direction of rotation the celtic stone about the vertical without any external action, and the origin of rotation in any direction due to oscillations about the horizontal axis.

The paper deals with the problem of integration of equations of motion in nonholonomic systems. By means of well-known theory of the differential equations with an invariant measure the new integrable systems are discovered. Among them there are the generalization of Chaplygin's problem of rolling nonsymmetric ball in the plane and the Suslov problem of rotation of rigid body with a fixed point. The structure of dynamics of systems on the invariant manifold in the integrable problems is shown. Some new ideas in the theory of integration of the equations in nonholonomic mechanics are suggested. The first of them consists in using known integrals as the constraints. The second is the use of resolvable groups of symmetries in nonholonomic systems. The existence conditions of invariant measure with analytical density for the differential equations of nonholonomic mechanics is given.

In this paper we study the cases of existence of an invariant measure, additional first integrals, and a Poisson structure in the problem of rigid body's rolling without sliding on a plane and a sphere. The problem of rigid body's motion on a plane was studied by S.A. Chaplygin, P. Appel, 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 the dynamically symmetrical body is a surface of revolution. These results were partially generalized by P. Woronetz, who studied the motion of a body of revolution and the motion of round disk with sharp edge on a sphere. In both cases the systems are Euler–Jacobi integrable and have additional integrals and invariant measure. It can be shown that by an appropriate change of time (determined by reducing multiplier), the reduced system is a Hamiltonian one. Here we consider some particular cases when the integrals and the invariant measure can be presented as finite algebraic expressions. We also consider a generalized problem of rolling of a dynamically nonsymmetric Chaplygin ball. The results of investigations are summarized in tables to illustrate the hierarchy of existence of various tensor invariants: invariant measure, integrals, and Poisson structure.

The paper is concerned with the problem on rolling of a homogeneous ball on an arbitrary surface. New cases when the problem is solved by quadratures are presented. The paper also indicates a special case when an additional integral and invariant measure exist. Using this case, we obtain a nonholonomic generalization of the Jacobi problem for the inertial motion of a point on an ellipsoid. For a ball rolling, it is also shown that on an arbitrary cylinder in the gravity field the ball's motion is bounded and, on the average, it does not move downwards. All the results of the paper considerably expand the results obtained by E. Routh in XIX century.

In this survey the basic concepts of nonholonomic systems such as equations of motion, the theory of reducing multiplier, variational principles and the Hamilton-Jacobi theorem are presented with some interesting bibliographical details.