Algebro-geometric Integration in Classical and Statistical Mechanics

Vladimir Dragovic

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.