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所属分类:数学  
出版时间:2009-1   出版时间:科学   作者:科兹洛夫   页数:184  

前言

  要使我国的数学事业更好地发展起来,需要数学家淡泊名利并付出更艰苦地努力。另一方面,我们也要从客观上为数学家创造更有利的发展数学事业的外部环境,这主要是加强对数学事业的支持与投资力度,使数学家有较好的工作与生活条件,其中也包括改善与加强数学的出版工作。  从出版方面来讲,除了较好较快地出版我们自己的成果外,引进国外的先进出版物无疑也是十分重要与必不可少的。从数学来说,施普林格(springer)出版社至今仍然是世界上最具权威的出版社。科学出版社影印一批他们出版的好的新书,使我国广大数学家能以较低的价格购买,特别是在边远地区工作的数学家能普遍见到这些书,无疑是对推动我国数学的科研与教学十分有益的事。  这次科学出版社购买了版权,一次影印了23本施普林格出版社出版的数学书,就是一件好事,也是值得继续做下去的事情。大体上分一下,这23本书中,包括基础数学书5本,应用数学书6本与计算数学书12本,其中有些书也具有交叉性质。这些书都是很新的,2000年以后出版的占绝大部分,共计16本,其余的也是1990年以后出版的。这些书可以使读者较快地了解数学某方面的前沿,例如基础数学中的数论、代数与拓扑三本,都是由该领域大数学家编著的“数学百科全书”的分册。对从事这方面研究的数学家了解该领域的前沿与全貌很有帮助。按照学科的特点,基础数学类的书以“经典”为主,应用和计算数学类的书以“前沿”为主。这些书的作者多数是国际知名的大数学家,例如《拓扑学》一书的作者诺维科夫是俄罗斯科学院的院士,曾获“菲尔兹奖”和“沃尔夫数学奖”。这些大数学家的著作无疑将会对我国的科研人员起到非常好的指导作用。  当然,23本书只能涵盖数学的一部分,所以,这项工作还应该继续做下去。更进一步,有些读者面较广的好书还应该翻译成中文出版,使之有更大的读者群。  总之,我对科学出版社影印施普林格出版社的部分数学著作这一举措表示热烈的支持,并盼望这一工作取得更大的成绩。

内容概要

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.

书籍目录

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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