This article is an enlarged version of the talk given by the author on the Meeting
on Mathematical Methods in Models of Mechanics, organized by Serbian Academy
of Sciences and Arts in its Novi Sad Branch in October 2003.The article is organized as follows. The next two sections contain necessary
notions and statements from algebraic geometry and integrable dynamical systems
- in Section 1 we list basic definitions related to the theory of integrable systems,
while Section 2 is a brief introduction to the theory of Riemann sUrfaces. In order
to keep the presentation reasonably short, we intensively assume two references
published in last few years in Belgrade [50, 26], and we refer readers to them
for details and clarifications regarding algebraic geometry and Poisson structures.
Let us emphasize that these mathematical techniques are the main tools for the
research performed in the framework of the Seminar on Mathematical Methods in
Mechanics.
In Section 3 we give a concise review of classical and modern results concerning
the motion of the rigid body about the fixed point. In Section 4, the original results
concerning a generalization of the classical Hess-Appel'rot rigid body system and
its integration in both classical and algebro-geometric ways are presented [22, 23].
In Section 5 we return again to classical subjects, presenting Poncelet theorem
on closed polygonal lines inscribed in one and circumscribed about another conic
in the plane and Cayley's condition that describe analytically such polygons. In
Section 6, billiards as an important class of dynamical systems are introduced.
In Section 7 , we present the original results - the generalization of the Cayley's
condition related to elliptical billiards in the space of arbitrary finite dimension
[27, 28]. Section 8 is aimed to present the author's results on separable potential
perturbations of integrable billiard systems [16, 17]. The last Section 9 is devoted to
exactly solvable models in Statistical Mechanics and problems of algebro-geometric
classification of the solutions of the Quantum Yang-Baxter equation.
