### Book

## Сборник контрольных работ и методических указаний для их выполнения по дисциплине «Методы математической физики». Методические указания к контрольным работам

In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum R -matrices. Here we study the simplest case – the 11-vertex R -matrix and related gl2 rational models. The corresponding top is equivalent to the 2-body Ruijsenaars–Schneider (RS) or the 2-body Calogero–Moser (CM) model depending on its description. We give different descriptions of the integrable tops and use them as building blocks for construction of more complicated integrable systems such as Gaudin models and classical spin chains (periodic and with boundaries). The known relation between the top and CM (or RS) models allows to rewrite the Gaudin models (or the spin chains) in the canonical variables. Then they assume the form of n -particle integrable systems with 2n constants. We also describe the generalization of the top to 1+1 field theories. It allows us to get the Landau–Lifshitz type equation. The latter can be treated as non-trivial deformation of the classical continuous Heisenberg model. In a similar way the deformation of the principal chiral model is described.

A Poisson structure is defined on the space W of twisted polygons in R^\nu. Poisson reductions with respect to two Poisson group actions on W are described. The \nu=2 and \nu=3 cases are discussed in detail. Amongst the Poisson structures arising in examples are to be found the lattice Virasoro structure, the second Toda lattice structure, and some extended Toda lattice structures. It is shown that, in general, for any \nu, the reduction procedure gives rise to a family of bi-Hamiltonian structures on the reduced space.

Authors developed algorithm of a complex assessment of efficiency of the advertising campaing