Vladimir Dragović
Full Research Professor of the Mathematical Institute of the Serbian Academy of Sciences and Arts
Head of the Department of Mechanics
The founder and chairman of the Dynamical Systems group
The cochairman of The Centre for Dynamical Systems, Geometry and Combinatorics of the Mathematical Institute of the Serbian Academy of Sciences and Arts
Born: July 26, 1967
1987: B.Sc. in Mathematics, University of Belgrade
19881992: aspirant at Moscow State University,Faculty of Mechanics and Mathematics,Department for Higher Geometry and Topology
1992: Dr. Sci. in Mathematics, University of Belgrade thesis:Rmatrices and algebraic curves advisor: Boris Dubrovin, Moscow State University
19922000, 20042007: teacher in Mathematical High School, Belgrade
19931998: Department of Mathematics, University of Belgrade, courses: Differential Geometry,several graduate courses
19961999: Head of the Committee for mathematical competitions of high school students of Serbia
19961999: Department of Philosophy, University of Nis courses: Differential Geometry, Partial Differential Equations
20032008: Department of Sciences and Mathematics, University of Montenegro courses: Geometry, Analysis and Geometry on Reimann Surfaces, Integrable Dynamical Systems 1, Integrable Dynamical Systems 2
20042008: regualar associate member of ICTP Abdus Salam, Trieste, Italy
20042008: Director of the Mathematical High School, Belgrade
20082012 Researcher, the Mathematical Physics Group, University of Lisbon
Employment
Since 2012, Full Professor of Mathematics, University of Texas at Dallas
Since 1988, employed at Mathematical Institute SANU
Conducting of scientific work
Chairman of the Seminar Mathematical Methods of Mechanics, since its founding in 1993
Advisor for four M.Sc. theses and four Ph.D. theses at Department of Mathematics, University of Belgrade
20012007, 2010: Member of the Committee for mathematics and mechanics of the Ministry for Sience of Serbia
20022005: Leader of Project 1643 of the Ministry for Sience of Serbia
since 2006: Leader of Project 144014 of the Ministry for Sience of Serbia
since 2005: CoLeader of the ItalianSerbian project Geometry, topology and combinatorics of manyfolds and dynamical systems
Awards
1987: Award for the best graduated student of the Faculty for Sciences and Mathematics
2004: Award of the Union of mathematical societies of Serbia and Montenegro for achievements in mathematical sciences for at most 40 years old researchers
2010: City of Belgrade Annual Award for natural and technical sciences (jointly with Milena Radnovic)
Publications:
Dragović V., Radnović M.
Billiards Within Ellipsoids in the 4Dimensional PseudoEuclidean Spaces
2023, vol. 28, no. 1, pp. 1443
Abstract
We study billiard systems within an ellipsoid in the 4dimensional pseudoEuclidean
spaces. We provide an analysis and description of periodic and weak periodic trajectories in
algebrogeometric and functionalpolynomial terms.

Dragović V., Gajić B., Jovanović B.
Spherical and Planar Ball Bearings — a Study of Integrable Cases
2023, vol. 28, no. 1, pp. 6277
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 wellknown 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.

Dragović V., Gajić B., Jovanović B.
Spherical and Planar Ball Bearings — Nonholonomic Systems with Invariant Measures
2022, vol. 27, no. 4, pp. 424442
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.

Dragović V., Gasiorek S., Radnović M.
Billiard Ordered Games and Books
2022, vol. 27, no. 2, pp. 132150
Abstract
The aim of this work is to put together two novel concepts from the theory of integrable billiards: billiard ordered games and confocal billiard books. Billiard books appeared recently in the work of Fomenko’s school, in particular, of V.Vedyushkina. These more complex billiard domains are obtained by gluing planar sets bounded by arcs of confocal conics along common edges. Such domains are used in this paper to construct the configuration space for billiard ordered games.We analyse dynamical and topological properties of the systems obtained in that way.

Adabrah A. K., Dragović V., Radnović M.
Periodic Billiards Within Conics in the Minkowski Plane and Akhiezer Polynomials
2019, vol. 24, no. 5, pp. 464501
Abstract
We derive necessary and sufficient conditions for periodic and for elliptic periodic
trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlining
elliptic curve. We provide several examples of periodic and elliptic periodic trajectories with
small periods. We observe a relationship between Cayleytype conditions and discriminantly
separable and factorizable polynomials. Equivalent conditions for periodicity and elliptic
periodicity are derived in terms of polynomialfunctional equations as well. The corresponding
polynomials are related to the classical extremal polynomials. In particular, the lightlike
periodic trajectories are related to the classical Chebyshev polynomials. Similarities and
differences with respect to the previously studied Euclidean case are highlighted.

Dragović V., Radnović M.
Caustics of Poncelet Polygons and Classical Extremal Polynomials
2019, vol. 24, no. 1, pp. 135
Abstract
A comprehensive analysis of periodic trajectories of billiards within ellipses in the Euclidean
plane is presented. The novelty of the approach is based on a relationship recently established
by the authors between periodic billiard trajectories and extremal polynomials on the systems
of $d$ intervals on the real line and ellipsoidal billiards in $d$dimensional space.
Even in the planar case systematically studied in the present paper, it leads to new results
in characterizing $n$ periodic trajectories vs. socalled $n$ elliptic periodic trajectories,
which are $n$periodic in elliptical coordinates. The characterizations are done both in terms
of the underlying elliptic curve and divisors on it and in terms of polynomial functional
equations, like Pell's equation. This new approach also sheds light on some classical results.
In particular, we connect the search for caustics which generate periodic trajectories with
three classical classes of extremal polynomials on two intervals, introduced by Zolotarev and Akhiezer.
The main classifying tool are winding numbers, for which we provide several interpretations, including
one in terms of numbers of points of alternance of extremal polynomials. The latter implies
important inequality between the winding numbers, which, as a consequence, provides another
proof of monotonicity of rotation numbers. A complete catalog of billiard trajectories with
small periods is provided for $n=3, 4, 5, 6$ along with an effective search for caustics.
As a byproduct, an intriguing connection between Cayleytype conditions and discriminantly
separable polynomials has been observed for all those small periods.

Dragović V., Gajić B., Jovanović B.
Note on Free Symmetric Rigid Body Motion
2015, vol. 20, no. 3, pp. 293308
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.

Dragović V., Gajić B.
FourDimensional Generalization of the Grioli Precession
2014, vol. 19, no. 6, pp. 656662
Abstract
A particular solution of the fourdimensional Lagrange top on $e(4)$ representing a fourdimensional regular precession is constructed. Using it, a fourdimensional analogue of the Grioli nonvertical regular precession of an asymmetric heavy rigid body is constructed.

Dragović V., Kukić K.
Systems of Kowalevski Type and Discriminantly Separable Polynomials
2014, vol. 19, no. 2, pp. 162184
Abstract
Starting from the notion of discriminantly separable polynomials of degree two in each of three variables, we construct a class of integrable dynamical systems. These systems can be integrated explicitly in genus two thetafunctions in a procedure which is similar to the classical one for the Kowalevski top. The discriminantly separable polynomials play the role of the Kowalevski fundamental equation. Natural examples include the Sokolov systems and the Jurdjevic elasticae.

Dragović V., Gajić B.
On the Cases of Kirchhoff and Chaplygin of the Kirchhoff Equations
2012, vol. 17, no. 5, pp. 431438
Abstract
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. Fourdimensional analogues of the Kirchhoff and the Chaplygin cases are defined on $e(4)$ with the standard Lie–Poisson bracket.

Dragović V., Kukić K.
New Examples of Systems of the Kowalevski Type
2011, vol. 16, no. 5, pp. 484495
Abstract
A new examples of integrable dynamical systems are constructed. An integration procedure leading to genus two thetafunctions is presented. It is based on a recent notion of discriminantly separable polynomials. They have appeared in a recent reconsideration of the celebrated Kowalevski top, and their role here is analogue to the situation with the classical Kowalevski integration procedure.

Dragović V., Gajić B.
Elliptic curves and a new construction of integrable systems
2009, vol. 14, no. 45, pp. 466478
Abstract
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 HessAppel’rot system is established. A sort of separation of variables is suggested for the HessAppel’rot system.

Dragović V., Radnović M.
Bifurcations of Liouville tori in elliptical billiards
2009, vol. 14, no. 45, pp. 479494
Abstract
A detailed description of topology of integrable billiard systems is given. For elliptical billiards and geodesic billiards on ellipsoid, the corresponding Fomenko graphs are constructed.

Dragović V., Gajić B.
Hirota–Kimura Type Discretization of the Classical Nonholonomic Suslov Problem
2008, vol. 13, no. 4, pp. 250256
Abstract
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 socalled steadystate rotations. The last property completely corresponds to wellknown property of the continuous Suslov case. The explicite formulae for solutions are given. In $n$dimensional case we give discrete equations.

Dragović V., Gajić B.
The Wagner Curvature Tenzor in Nonholonomic Mechanics
2003, vol. 8, no. 1, pp. 105123
Abstract
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.
