Božidar Jovanović

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.

In 1983 Bogoyavlenski conjectured that, if the Euler equations on a Lie algebra $\mathfrak{g}_0$ are integrable, then their certain extensions to semisimple lie algebras $\mathfrak{g}$ related to the filtrations of Lie algebras $\mathfrak{g}_0\subset\mathfrak{g}_1\subset\mathfrak{g}_2\dots\subset\mathfrak{g}_{n-1}\subset \mathfrak{g}_n=\mathfrak{g}$ are integrable as well. In particular, by taking $\mathfrak{g}_0=\{0\}$ and natural filtrations of ${\mathfrak{so}}(n)$ and $\mathfrak{u}(n)$, we have Gel’fand – Cetlin integrable systems. We prove the conjecture for filtrations of compact Lie algebras $\mathfrak{g}$: the system is integrable in a noncommutative sense by means of polynomial integrals. Various constructions of complete commutative polynomial integrals for the system are also given.

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.

In the review we describe a relation between the Heisenberg spin chain model on pseudospheres and light-like cones in pseudo-Euclidean spaces and virtual billiards. A geometrical interpretation of the integrals associated to a family of confocal quadrics is given, analogous to Moser’s geometrical interpretation of the integrals of the Neumann system on the sphere.

We introduce a class of dynamical systems having an invariant measure, the modifications of well-known systems on Lie groups: LR and L$+$R systems. As an example, we study a modified Veselova nonholonomic rigid body problem, considered as a dynamical system on the product of the Lie algebra $so(n)$ with the Stiefel variety $V_{n,r}$, as well as the associated $\epsilon$L$+$R system on $so(n)\times V_{n,r}$. In the 3-dimensional case, these systems model the nonholonomic problems of motion of a ball and a rubber ball over a fixed sphere.

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.

We construct analogues of the classical Heisenberg spin chain model (or the discrete Neumann system), on pseudo-spheres and light-like cones in the pseudo-Euclidean spaces and show their complete Hamiltonian integrability. Further, we prove that the Heisenberg model on a light-like cone leads to a new example of the integrable discrete contact system.

We prove the integrability of geodesic flows on the Riemannian g.o. spaces of compact Lie groups, as well as on a related class of Riemannian homogeneous spaces having an additional principal bundle structure.

We revise the solution to the problem of Hamiltonization of the $n$-dimensional Veselova non-holonomic system studied previously in [1]. Namely, we give a short and direct proof of the hamiltonization theorem and also show the trajectorial equivalence of the problem with the geodesic flow on the ellipsoid.

In this paper we study the dynamics of the constrained $n$-dimensional rigid body (the Suslov problem). We give a review of known integrable cases in three dimensions and present their higher dimensional generalizations.