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2013
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# Božidar Jovanović

Kneza Mihaila 35, 11000 Belgrade, Serbia
Mathematical Institute, SANU

## Publications:

 Dragović V., Gajić B., Jovanović B. Spherical and Planar Ball Bearings — a Study of Integrable Cases 2023, vol. 28, no. 1, pp.  62-77 Abstract 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. Keywords: nonholonimic dynamics, rolling without slipping, invariant measure, integrability Citation: Dragović V., Gajić B., Jovanović B.,  Spherical and Planar Ball Bearings — a Study of Integrable Cases, Regular and Chaotic Dynamics, 2023, vol. 28, no. 1, pp. 62-77 DOI:10.1134/S1560354723010057
 Jovanović B., Šukilović  T., Vukmirović S. Integrable Systems Associated to the Filtrations of Lie Algebras 2023, vol. 28, no. 1, pp.  44-61 Abstract 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. Keywords: noncommutative integrability, invariant polynomials, Gel’fand – Cetlin systems Citation: Jovanović B., Šukilović  T., Vukmirović S.,  Integrable Systems Associated to the Filtrations of Lie Algebras, Regular and Chaotic Dynamics, 2023, vol. 28, no. 1, pp. 44-61 DOI:10.1134/S1560354723010045
 Dragović V., Gajić B., Jovanović B. Spherical and Planar Ball Bearings — Nonholonomic Systems with Invariant Measures 2022, vol. 27, no. 4, pp.  424-442 Abstract 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. Keywords: nonholonimic dynamics, rolling without slipping, invariant measure, integrability Citation: Dragović V., Gajić B., Jovanović B.,  Spherical and Planar Ball Bearings — Nonholonomic Systems with Invariant Measures, Regular and Chaotic Dynamics, 2022, vol. 27, no. 4, pp. 424-442 DOI:10.1134/S1560354722040037
 Jovanović B., Jovanovic V. Heisenberg Model in Pseudo-Euclidean Spaces II 2018, vol. 23, no. 4, pp.  418-437 Abstract 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. Keywords: discrete systems with constraints, contact integrability, billiards, Neumann and Heisenberg systems Citation: Jovanović B., Jovanovic V.,  Heisenberg Model in Pseudo-Euclidean Spaces II, Regular and Chaotic Dynamics, 2018, vol. 23, no. 4, pp. 418-437 DOI:10.1134/S1560354718040044
 Jovanović B. Invariant Measures of Modified LR and L$+$R Systems 2015, vol. 20, no. 5, pp.  542-552 Abstract 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. Keywords: nonholonomic constraints, invariant measure, Chaplygin ball Citation: Jovanović B.,  Invariant Measures of Modified LR and L$+$R Systems, Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 542-552 DOI:10.1134/S1560354715050032
 Dragović V., Gajić B., Jovanović B. Note on Free Symmetric Rigid Body Motion 2015, vol. 20, no. 3, pp.  293-308 Abstract 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. Keywords: Euler equations, Manakov integrals, spectral curve, reduced Poisson space Citation: Dragović V., Gajić B., Jovanović B.,  Note on Free Symmetric Rigid Body Motion, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 293-308 DOI:10.1134/S1560354715030065
 Jovanović B. Heisenberg Model in Pseudo-Euclidean Spaces 2014, vol. 19, no. 2, pp.  245-250 Abstract 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. Keywords: discrete Hamiltonian and contact systems, the Lax representation, complete integrability Citation: Jovanović B.,  Heisenberg Model in Pseudo-Euclidean Spaces, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 245-250 DOI:10.1134/S1560354714020075
 Jovanović B. Geodesic Flows on Riemannian g.o. Spaces 2011, vol. 16, no. 5, pp.  504-513 Abstract 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. Keywords: noncommutative integrability, geodesic orbit spaces, complexity of homogeneous spaces, fiber bundles Citation: Jovanović B.,  Geodesic Flows on Riemannian g.o. Spaces, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 504-513 DOI:10.1134/S1560354711050078
 Fedorov Y. N., Jovanović B. Hamiltonization of the generalized Veselova LR system 2009, vol. 14, no. 4-5, pp.  495-505 Abstract 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. Keywords: nonholonomic systems, integrability, geodesic flows Citation: Fedorov Y. N., Jovanović B.,  Hamiltonization of the generalized Veselova LR system, Regular and Chaotic Dynamics, 2009, vol. 14, no. 4-5, pp. 495-505 DOI:10.1134/S1560354709040066
 Jovanović B. Some Multidimensional Integrable Cases Of Nonholonomic Rigid Body Dynamics 2003, vol. 8, no. 1, pp.  125-132 Abstract 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. Citation: Jovanović B.,  Some Multidimensional Integrable Cases Of Nonholonomic Rigid Body Dynamics, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 125-132 DOI:10.1070/RD2003v008n01ABEH000230