動(dòng)力系統(tǒng)X

前言

　　要使我國(guó)的數(shù)學(xué)事業(yè)更好地發(fā)展起來(lái)，需要數(shù)學(xué)家淡泊名利并付出更艱苦地努力。另一方面，我們也要從客觀上為數(shù)學(xué)家創(chuàng)造更有利的發(fā)展數(shù)學(xué)事業(yè)的外部環(huán)境，這主要是加強(qiáng)對(duì)數(shù)學(xué)事業(yè)的支持與投資力度，使數(shù)學(xué)家有較好的工作與生活條件，其中也包括改善與加強(qiáng)數(shù)學(xué)的出版工作。　　從出版方面來(lái)講，除了較好較快地出版我們自己的成果外，引進(jìn)國(guó)外的先進(jìn)出版物無(wú)疑也是十分重要與必不可少的。從數(shù)學(xué)來(lái)說(shuō)，施普林格（springer）出版社至今仍然是世界上最具權(quán)威的出版社?？茖W(xué)出版社影印一批他們出版的好的新書(shū)，使我國(guó)廣大數(shù)學(xué)家能以較低的價(jià)格購(gòu)買(mǎi)，特別是在邊遠(yuǎn)地區(qū)工作的數(shù)學(xué)家能普遍見(jiàn)到這些書(shū)，無(wú)疑是對(duì)推動(dòng)我國(guó)數(shù)學(xué)的科研與教學(xué)十分有益的事?！　∵@次科學(xué)出版社購(gòu)買(mǎi)了版權(quán)，一次影印了23本施普林格出版社出版的數(shù)學(xué)書(shū)，就是一件好事，也是值得繼續(xù)做下去的事情。大體上分一下，這23本書(shū)中，包括基礎(chǔ)數(shù)學(xué)書(shū)5本，應(yīng)用數(shù)學(xué)書(shū)6本與計(jì)算數(shù)學(xué)書(shū)12本，其中有些書(shū)也具有交叉性質(zhì)。這些書(shū)都是很新的，2000年以后出版的占絕大部分，共計(jì)16本，其余的也是1990年以后出版的。這些書(shū)可以使讀者較快地了解數(shù)學(xué)某方面的前沿，例如基礎(chǔ)數(shù)學(xué)中的數(shù)論、代數(shù)與拓?fù)淙?，都是由該領(lǐng)域大數(shù)學(xué)家編著的“數(shù)學(xué)百科全書(shū)”的分冊(cè)。對(duì)從事這方面研究的數(shù)學(xué)家了解該領(lǐng)域的前沿與全貌很有幫助。按照學(xué)科的特點(diǎn)，基礎(chǔ)數(shù)學(xué)類的書(shū)以“經(jīng)典”為主，應(yīng)用和計(jì)算數(shù)學(xué)類的書(shū)以“前沿”為主。這些書(shū)的作者多數(shù)是國(guó)際知名的大數(shù)學(xué)家，例如《拓?fù)鋵W(xué)》一書(shū)的作者諾維科夫是俄羅斯科學(xué)院的院士，曾獲“菲爾茲獎(jiǎng)”和“沃爾夫數(shù)學(xué)獎(jiǎng)”。這些大數(shù)學(xué)家的著作無(wú)疑將會(huì)對(duì)我國(guó)的科研人員起到非常好的指導(dǎo)作用?！　‘?dāng)然，23本書(shū)只能涵蓋數(shù)學(xué)的一部分，所以，這項(xiàng)工作還應(yīng)該繼續(xù)做下去。更進(jìn)一步，有些讀者面較廣的好書(shū)還應(yīng)該翻譯成中文出版，使之有更大的讀者群?！　】傊覍?duì)科學(xué)出版社影印施普林格出版社的部分?jǐn)?shù)學(xué)著作這一舉措表示熱烈的支持，并盼望這一工作取得更大的成績(jī)。

內(nèi)容概要

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.

書(shū)籍目錄

IntroductionDescartes, Leibnitz, and NewtonNewton and BernoulliVoltaire, Maupertuis, and ClairautHelmholtz and ThomsonAbout the BookChapter 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 ProblemChapter 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 SystemsChapter 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 ManifoldsChapter 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 FreedomSupplement 1: Vorticity Invariants and Secondary HydrodynamicsSupplement 2: Quantum Mechanics and HydrodynamicsSupplement 3: Vortex Theory of Adiabatic Equilibrium ProcessesReferencesIndex

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