This book is a collection of the most significant classical results on the dynamics of liquid and gaseous ellipsoids, starting with the fundamental investigations of Dirichlet and Riemann. The papers of the collection deal primarily with the derivation of various forms of the equations of motion and the investigation of qualitative properties of the dynamics of ellipsoidal figures. The book addresses specialists and graduate students interested in mechanics, mathematical physics and the history of science.

This book is a collection of the main classical works concerned with the Clebsch problem and related integrable systems. These papers, written by outstanding mathematicians of XIX centry, had a substantial influence on the development of many fields of modern mathematics and physics. The study of the Clebsch case and equivalent systems is far from completion, therefore, this problem remains one of the central in the theory of integrable systems. The book addresses researchers, undergraduate and graduate students interested in theoretical mechanics, mathematical physics and the history of science.

This work brings together previously unpublished notes contributed by participants of the IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence held in Moscow, August 25-30, 2006.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.

In this book we discuss the main forms of equations of motion of rigid-body systems, such as motion of rigid bodies in potential fields, in fluid (Kirchhoff equation) and rigid bodies with fluid-filled cavities. All the systems considered in the book can be described within the framework of the Hamiltonian formalism. Almost all known integrable cases and methods of explicit integration are included. Compared to the previous volume, new sections dealing with non-integrability analysis and chaos in various problems of rigid body dynamics are added. Computer-based visualization of motion is widely used. Some results are obtained by the authors.

The book describes the main mathematical methods of investigation of vortex structures in an ideal incompressible fluid. All the methods of analysis of integrability and non-integrable systems are based on a systematic use of the Hamiltonian formalism and qualitative analysis. Some topics discussed in the book are: motion of point vortices on a plane and a sphere, interaction of vortex patches and some fresh issues concerned with dynamical interaction between rigid bodies and vortex structures in an ideal fluid. The appendices contain some new results, obtained by the authors in cooperation with their students and colleagues.

The book is a collecton of recent and classical works of dynamics in spaces of constant curvature. The Kepler problem and its extentions, the two- and three-body problem and the rigid body dinamics in curved spaces are considered. Many classical works by W. Killing, H.Liebmann, etc. have been practically not available for a wide audience and almost forgotten. The recent papers collected here discuss stochasticity and integrability issues, various generalizations of some results from classical and celestial mechanics, and the Newton potential theory. The book is for undergraduate and postgraduate students and specialists in dynamical systems. We hope that it will be interesting for scientific historians.

The book includes works of national and abroad authors examining the dynamics of vortex structures in fluid. The collected articles show this domain of study as an intensively evolving branch of fluid dynamics. With this aim in view, there are given both the results that have become well known, and the last achievements of the authors. The first part of the book is dedicated to the statement and solution of problems formulated in the frames of the classic hydrodynamic theory. Vortex problems of the geophysical hydrodynamics are stated in the second part of the book. This collection of articles will be useful for the specialists in the field of dynamic systems and hydrodynamics, for lecturers, post-graduates and students in this scientific branch.

The book studies integrable systems of the Hamiltonian mechanics within the context of the Lax representation and the explicit integration procedures. The authors introduce new methods of separation of variables and formulate the universal algorithm for constructing L-A pairs based on bi-hamiltonianity. In the book are also discussed multidimensional analogues of the integrable problems in the rigid body dynamics, generalized Toda lattices, geodesic flows and other problems in mechanics and geometry.

The book is a collection of papers concerned with dynamical effects in the motion of nonholonomic systems. Most of the contributions have been exclusively written for this book by leading Russian experts and involve novel results. These include, among others, new geometrical images of dynamics and various hierarchies of systems behavior. Also three-dimensional mappings in the problems of rolling motion of bodies are investigated numerically.

Here the fundamental forms of the equations of motion for rigid body are discussed. These involve motion in potential fields and in fluid (the Kirchhoff equations), motion of a body with cavities filled with fluid. Conditions under which the reduction of the order of these equations is possible and cyclic variables exist are given. In addition, the book collects almost all known to date integrable cases along with methods of their explicit integration. Throughout the book results of computer simulations are copiously presented to help visualize motions' peculiarities. Most results discussed in the book are obtained by the authors.

The book is about Poisson structures and their application to various problems of Hamiltonian mechanics arising in a number of areas, such as dynamics of rigid body, celestial mechanics, the theory of vortices, cosmological models. The equations governing the motion of such systems can be written in a convenient polynomial (algebraic) form. This form is in close connection with the possibility of representing the equations of motion as a set of Hamiltonian equations with linear Poisson structure associated with some Lie algebra. The authors also discuss nonlinear Poisson structures defined by infinite-dimensional Lie algebras and consider most typical situations in which such structures occur. The equations obtained are studied using the Painleve–Kovalevskaya method. Also the book presents some new integrable cases and establishes isomorphisms between integrable problems.

In this paper we present analytical and numerical data on nonintegrability of the equation of motion and the stochastic behaviour of trajectories in a range of classical and modern problems of dynamics of rigid body. There are considered analytical methods of proof of non-integrability in the systems close to integrable and their relation with data of numerical simulation. The observed spliting of separatrixes leads to absence of the additional first integrals and formation of stochastic layers within chaos in dynamics of rigid body.