Dynamical Systems X

1. Front Matter

   Pages I-VIII

   PDF

2. Introduction

   • V. V. Kozlov

   Pages 1-8

3. Hydrodynamics, Geometric Optics, and Classical Mechanics

   • V. V. Kozlov

   Pages 9-75

4. General Vortex Theory

   • V. V. Kozlov

   Pages 76-107

5. Geodesics on Lie Groups with a Left-Invariant Metric

   • V. V. Kozlov

   Pages 108-134

6. Vortex Method for Integrating Hamilton Equations

   • V. V. Kozlov

   Pages 135-159

7. Back Matter

   Pages 160-184

   PDF

The English teach mechanics as an experimental science, while on the Continent, it has always been considered a more deductive and a priori science. Unquestionably, the English are right. * H. Poincare, Science and Hypothesis Descartes, Leibnitz, and Newton As is well known, the basic principles of dynamics were stated by New­ ton in his famous work Philosophiae Naturalis Principia Mathematica, whose publication in 1687 was paid for by his friend, the astronomer Halley. In essence, this book was written with a single purpose: to prove the equivalence of Kepler's laws and the assumption, suggested to Newton by Hooke, that the acceleration of a planet is directed toward the center of the Sun and decreases in inverse proportion to the square of the distance between the planet and the Sun. For this, Newton needed to systematize the principles of dynamics (which is how Newton's famous laws appeared) and to state the "theory of fluxes" (analysis of functions of one variable). The principle of the equality of an action and a counteraction and the inverse square law led Newton to the theory of gravitation, the interaction at a distance. In addition, New­ ton discussed a large number of problems in mechanics and mathematics in his book, such as the laws of similarity, the theory of impact, special vari­ ational problems, and algebraicity conditions for Abelian integrals. Almost everything in the Principia subsequently became classic. In this connection, A. N.

• Hamilton equations
• Hamilton-Jacobi method
• Stability theory
• classical mechanics
• hydrodynamics
• quantum mechanics
• vortex

From the reviews:

"This is a very interesting, nicely written book on deep mathematical analogies between hydrodynamics, geometric optics and mechanics. […] The book will surely be of interest to researchers and postgraduate students in mathematical physics and mechanics."

EMS Newsletter December 2005

"... Kozlov's very valuable contribution to the Encyclopaedia of Mathematical Sciences explores the world from the point of view of the vortex, its role in a variety of theories with an acknowledgement of the history of the argument originating in the rivalry of the Cartesian and Newtonian views of the world. ..."

C.Athorne, University of Glasgow, Contemporary Physics 2004, Vol. 45, Issue 5

"Kozlov’s very valuable contribution to the Encyclopedia of Mathematical Sciences explores the world from the point of view of the vortex, its role in a variety of theories with an acknowledgement of the history of the argument originating in the rivalry of the Cartesian and Newtonian views of the world. Thus, the impulse behind the book is to make precise and formalize analogies between the motions of ideal fluids and the classical mechanics of conservative systems."

Dr. C. Athorne, Contemporary Physics, Vol. 45 (5), 2004

"The book is a translation from the 1998 original work appeared in Russian. … the author highlights some analogies between hydrodynamics, classical mechanics, geometrical optics, and symplectic geometry. … The book will be of great interest to researchers working on this or related subjects."

G.Schneider, ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 84 (10-11), 2004

"This book is devoted to a mathematical description of analogies between hydrodynamics, geometrical optics and classical mechanics. The basic idea is that the study of families of trajectories of Hamiltonian systems can be reduced to the investigation of multidimensionalideal fluid dynamics. The book contains four chapters and three appendices. … can be recommended to researchers and graduate students interested in mathematical physics, mechanics and differential equations." (Tibor Krisztin, Acta Scientiarum Mathematicarum, Vol. 74, 2008)

• Department of Mathematics and Mechanics, Moscow State University, Moscow, Russia

  V. V. Kozlov

Policies and ethics