Milena Radnović
Kneza Mihaila 36, 11001, Belgrade, p.p. 367, Serbia
Mathematical Institute SANU
Publications:
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., 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.

Radnović M., RomKedar V.
Foliations of isonergy surfaces and singularities of curves
2008, vol. 13, no. 6, pp. 645668
Abstract
It is well known that changes in the Liouville foliations of the isoenergy surfaces of an integrable system imply that the bifurcation set has singularities at the corresponding energy level.We formulate certain genericity assumptions for two degrees of freedom integrable systems and we prove the opposite statement: the essential critical points of the bifurcation set appear only if the Liouville foliations of the isoenergy surfaces change at the corresponding energy levels. Along the proof, we give full classification of the structure of the isoenergy surfaces near the critical set under our genericity assumptions and we give their complete list using Fomenko graphs. This may be viewed as a step towards completing the Smale program for relating the energy surfaces foliation structure to singularities of the momentum mappings for nondegenerate integrable two degrees of freedom systems.
