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《国外数学名著系列(续1)(影印版)54:动力系统10(旋涡的一般理论》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.

IntroductionDescartes, Leibnitz, and NewtonNewton and BernoulliVoltaire, Maupertuis, and ClairautHelmholtz and ThomsonAbout the BookChapter 1. Hydrodynamics, Geometric Optics, and Classical Mechanics1. Vortex Motions of a Continuous Medium2. Point Vortices on the Plane3. Systems of Rays, Laws of Reflection and Refraction, and the Malus Theorem4. Fermat Principle, Canonical Hamilton Equations, and the Optical-Mechanical Analogy5. Hamiltonian Form of the Equations of Motion6. Action in the Phase Space and the Poincare-Cartan Invariant7. Hamilton-Jacobi Method and Huygens Principle8. Hydrodynamics of Hamiltonian Systems9. Lamb Equations and the Stability ProblemChapter 2. General Vortex Theory1. Lamb Equations and Hamilton Equations2. Reduction to the Autonomous Case3. Invariant Volume Forms4. Vortex Manifolds5. Euler Equation6. Vortices in Dissipative SystemsChapter 3. Geodesics on Lie Groups with a Left-Invariant Metric1. Euter-Poincare Equations2. Vortex Theory of the Top3. Haar Measure4. Poisson Brackets5. Casimir Functions and Vortex ManifoldsChapter 4. Vortex Method for Integrating Hamilton Equations1. Hamilton-Jacobi Method and the Liouville Theorem on Complete Integrability2. Noncommutative Integration of the Hamilton Equations3. Vortex Integration Method4. Complete Integrability of the Quotient System5. Systems with Three Degrees of FreedomSupplement 1: Vorticity Invariants and Secondary HydrodynamicsSupplement 2: Quantum Mechanics and HydrodynamicsSupplement 3: Vortex Theory of Adiabatic Equilibrium ProcessesReferencesIndex

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