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商品簡介
目次
商品簡介
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.
目次
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
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
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