Borislav Gajić

We consider the nonholonomic systems of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with the same radius $r$ that are rolling without slipping about a fixed sphere $\mathbf S_0$ with center $O$ and radius $R$. In addition, it is assumed that a dynamically nonsymmetric sphere $\mathbf S$ with the center that coincides with the center $O$ of the fixed sphere $\mathbf S_0$ rolls without slipping in contact with the moving balls $\mathbf B_1,\dots,\mathbf B_n$. The problem is considered in four different configurations, three of which are new. We derive the equations of motion and find an invariant measure for these systems. As the main result, for $n=1$ we find two cases that are integrable by quadratures according to the Euler – Jacobi theorem. The obtained integrable nonholonomic models are natural extensions of the well-known Chaplygin ball integrable problems. Further, we explicitly integrate the planar problem consisting of $n$ homogeneous balls of the same radius, but with different masses, which roll without slipping over a fixed plane $\Sigma_0$ with a plane $\Sigma$ that moves without slipping over these balls.

We first construct nonholonomic systems of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with centers $O_1,\ldots,O_n$ and with the same radius $r$ that are rolling without slipping around a fixed sphere $\mathbf S_0$ with center $O$ and radius $R$. In addition, it is assumed that a dynamically nonsymmetric sphere $\mathbf S$ of radius $R+2r$ and the center that coincides with the center $O$ of the fixed sphere $\mathbf S_0$ rolls without slipping over the moving balls $\mathbf B_1,\dots,\mathbf B_n$. We prove that these systems possess an invariant measure. As the second task, we consider the limit, when the radius $R$ tends to infinity. We obtain a corresponding planar problem consisting of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with centers $O_1,\ldots,O_n$ and the same radius $r$ that are rolling without slipping over a fixed plane $\Sigma_0$, and a moving plane $\Sigma$ that moves without slipping over the homogeneous balls. We prove that this system possesses an invariant measure and that it is integrable in quadratures according to the Euler – Jacobi theorem.

We consider the Euler equations of motion of a free symmetric rigid body around a fixed point, restricted to the invariant subspace given by the zero values of the corresponding linear Noether integrals. In the case of the $SO(n − 2)$-symmetry, we show that almost all trajectories are periodic and that the motion can be expressed in terms of elliptic functions. In the case of the $SO(n − 3)$-symmetry, we prove the solvability of the problem by using a recent Kozlov’s result on the Euler–Jacobi–Lie theorem.

A particular solution of the four-dimensional Lagrange top on $e(4)$ representing a four-dimensional regular precession is constructed. Using it, a four-dimensional analogue of the Grioli nonvertical regular precession of an asymmetric heavy rigid body is constructed.

It is proven that the completely integrable general Kirchhoff case of the Kirchhoff equations for $B \ne 0$ is not an algebraic complete integrable system. Similar analytic behavior of the general solution of the Chaplygin case is detected. Four-dimensional analogues of the Kirchhoff and the Chaplygin cases are defined on $e(4)$ with the standard Lie–Poisson bracket.

A class of elliptic curves with associated Lax matrices is considered. A family of dynamical systems on $e(3)$ parametrized by polynomial a with the above Lax matrices are constructed. Five cases from the family are selected by the condition of preserving the standard measure. Three of them are Hamiltonian. It is proved that two other cases are not Hamiltonian in the standard Poisson structure on $e(3)$. Integrability of all five cases is proven. Integration procedures are performed in all five cases. Separation of variables in Sklyanin sense is also given. A connection with Hess-Appel’rot system is established. A sort of separation of variables is suggested for the Hess-Appel’rot system.

We constructed Hirota–Kimura type discretization of the classical nonholonomic Suslov problem of motion of rigid body fixed at a point. We found a first integral proving integrability. Also, we have shown that discrete trajectories asymptotically tend to a line of discrete analogies of so-called steady-state rotations. The last property completely corresponds to well-known property of the continuous Suslov case. The explicite formulae for solutions are given. In $n$-dimensional case we give discrete equations.

We present the classical Wagner construction from 1935 of the curvature tensor for the completely nonholonomic manifolds in both invariant and coordinate way. The starting point is the Shouten curvature tensor for the nonholonomic connection introduced by Vranceanu and Shouten. We illustrate the construction by two mechanical examples: the case of a homogeneous disc rolling without sliding on a horizontal plane and the case of a homogeneous ball rolling without sliding on a fixed sphere. In the second case we study the conditions imposed on the ratio of diameters of the ball and the sphere to obtain a flat space — with the Wagner curvature tensor equal to zero.