This work brings together previously unpublished notes contributed by participants of the IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence held in Moscow, August 25-30, 2006. The study of vortex motion is of great interest to fluid and gas dynamics; since all real flows are vortical in nature, applications of the vortex theory are extremely diverse, many of them (e.g. aircraft dynamics, atmospheric and ocean phenomena) being especially important. The last few decades have shown that serious possibilities for progress in the research of real turbulent vortex motions are essentially related to the combined use of mathematical methods, computer simulation and laboratory experiments. These approaches have led to a series of interesting results which allow us to study these processes from new perspectives. Based on this principle, the papers collected in this proceedings volume present new results on theoretical and applied aspects and processes of formation and evolution of various flows, wave and coherent structures in gas and fluid. Much attention is given to the studies of nonlinear regular and chaotic regimes of vortex interactions, advective and convective motions. The contributors are leading scientists engaged in fundamental and applied aspects of the above mentioned fields.

According to Stephan Banach, the possibilities of a mathematician are determined by his abilities to operate with analogies of different levels. The first level is the ability to see and to use the analogies between different problems. The next level is constituted by the analogies between theories. And finally only can be considered as high class mathematicians those who are able to use analogies between analogies.

Billiard systems arise in many problems of mechanics and physics, some of them being described in the introduction. The authors have chosen a genetic approach, which consists in replacing a single-sided constraint imposed on the system by a field of elastic and dissipative forces. When the elastic and dissipative coefficients tend to infinity, the motion with fixed initial values tends on finite time intervals to a motion with impacts. The celebrated theorem of Birkhoff on the estimating of the number of distinct periodic trajectories of a convex elastic billiard is proved in a variational way, and some conditions for stability of the periodic trajectories are given. A chapter is dedicated to the Hill equation, obtained in the problem of the motion of the perigee of the moon and encountered also in problems of the stability of periodic motions with impacts. An important problem is the integrability of systems of billiard type. Old and new integrable problems, as the elliptic billiard, the billiard in affine Weyl cells, a harmonic oscillator inside an ellipse, billiards on surfaces of constant curvatures are described. The book is written in a clear and careful style and completely describes the problems from the physical phenomenon to the mathematical solution.