This book contains a mathematical exposition of analogies between classical (Hamiltonian) mechanics, geometrical optics, and hydrodynamics. This theory highlights several general mathematical ideas that appeared in Hamiltonian mechanics, optics and hydrodynamics under different names. In addition, some interesting applications of the general theory of vortices are discussed in the book such as applications in numerical methods, stability theory, and the theory of exact integration of equations of dynamics. The investigation of families of trajectories of Hamiltonian systems can be reduced to problems of multidimensional ideal fluid dynamics.For example, the well-known Hamilton-Jacobi method corresponds to the case of potential flows. The book will be of great interest to researchers and postgraduate students interested in mathematical physics, mechanics, and the theory of differential equations.

Introduction

Descartes, Leibnitz, and Newton

Newton and Bernoulli

Voltaire, Maupertuis, and Clairaut

Helmholtz and Thomson

About the Book

Chapter 1. Hydrodynamics, Geometric Optics, and Classical Mechanics

1. Vortex Motions of a Continuous Medium

2. Point Vortices on the Plane

3. Systems of Rays, Laws of Reflection and Refraction, and the Malus Theorem

4. Fermat Principle, Canonical Hamilton Equations, and the Optical-Mechanical Analogy

5. Hamiltonian Form of the Equations of Motion

6. Action in the Phase Space and the Poincare-Cartan Invariant

7. Hamilton-Jacobi Method and Huygens Principle

8. Hydrodynamics of Hamiltonian Systems

9. Lamb Equations and the Stability Problem

Chapter 2. General Vortex Theory

1. Lamb Equations and Hamilton Equations

2. Reduction to the Autonomous Case

3. Invariant Volume Forms

4. Vortex Manifolds

5. Euler Equation

6. Vortices in Dissipative Systems

Chapter 3. Geodesics on Lie Groups with a Left-Invariant Metric

1. Euter-Poincare Equations

2. Vortex Theory of the Top

3. Haar Measure

4. Poisson Brackets

5. Casimir Functions and Vortex Manifolds

Chapter 4. Vortex Method for Integrating Hamilton Equations

1. Hamilton-Jacobi Method and the Liouville Theorem on Complete Integrability

2. Noncommutative Integration of the Hamilton Equations

3. Vortex Integration Method

4. Complete Integrability of the Quotient System

5. Systems with Three Degrees of Freedom

Supplement 1: Vorticity Invariants and Secondary Hydrodynamics

Supplement 2: Quantum Mechanics and Hydrodynamics

Supplement 3: Vortex Theory of Adiabatic Equilibrium Processes

References

Index