經典力學與天體力學中的數學問題

前言

要使我國的數學事業(yè)更好地發(fā)展起來，需要數學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學家創(chuàng)造更有利的發(fā)展數學事業(yè)的外部環(huán)境，這主要是加強對數學事業(yè)的支持與投資力度，使數學家有較好的工作與生活條件，其中也包括改善與加強數學的出版工作。從出版方面來講，除了較好較快地出版我們自己的成果外，引進國外的先進出版物無疑也是十分重要與必不可少的。從數學來說，施普林格（Springer）出版社至今仍然是世界上最具權威的出版社?？茖W出版社影印一批他們出版的好的新書，使我國廣大數學家能以較低的價格購買，特別是在邊遠地區(qū)工作的數學家能普遍見到這些書，無疑是對推動我國數學的科研與教學十分有益的事。這次科學出版社購買了版權，一次影印了23本施普林格出版社出版的數學書，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本書中，包括基礎數學書5本，應用數學書6本與計算數學書12本，其中有些書也具有交叉性質。這些書都是很新的，2000年以后出版的占絕大部分，共計16本，其余的也是1990年以后出版的。這些書可以使讀者較快地了解數學某方面的前沿，例如基礎數學中的數論、代數與拓撲三本，都是由該領域大數學家編著的“數學百科全書”的分冊。對從事這方面研究的數學家了解該領域的前沿與全貌很有幫助。按照學科的特點，基礎數學類的書以“經典”為主，應用和計算數學類的書以“前沿”為主。這些書的作者多數是國際知名的大數學家，例如《拓撲學》一書的作者諾維科夫是俄羅斯科學院的院士，曾獲“菲爾茲獎”和“沃爾夫數學獎”。這些大數學家的著作無疑將會對我國的科研人員起到非常好的指導作用。

內容概要

This work describes the fundamental principles, problems, and methods of classical mechanics. The main attention is devoted to the mathematical side of the subject. The authors have endeavored to give an exposition stressing the working apparatus of classical mechanics. The book is significantly expanded compared to the previous edition. The authors have added two chapters on the variational principles and methods of classical mechanics as well as on tensor invariants of equations of dynamics. Moreover, various other sections have been revised, added or expanded. The main purpose of the book is to acquaint the reader with classical mechanics as a whole, in both its classical and its contemporary aspects.The book addresses all mathematicians, physicists and engineers.

作者簡介

作者：（俄羅斯）阿諾德（Vladimir I.Arnold） 等

書籍目錄

1 Basic Principles of Classical Mechanics  1.1  Newtonian Mechanics    1.1.1  Space, Time, Motion    1.1.2  Newton-Laplace Principle of Determinacy    1.1.3  Principle of Relativity    1.1.4  Principle of Relativity and Forces of Inertia    1.1.5  Basic Dynamical Quantities. Conservation Laws...  1.2  Lagrangian Mechanics    1.2.1  Preliminary Remarks    1.2.2  Variations and Extremals    1.2.3  Lagrange's Equations    1.2.4  Poincare's Equations    1.2.5  Motion with Constraints  1.3  Hamiltonian Mechanics    1.3.1  Symplectic Structures and Hamilton's Equations     1.3.2  Generating Functions    1.3.3  Symplectic Structure of the Cotangent Bundle     1.3.4  The Problem of n Point Vortices    1.3.5  Action in the Phase Space    1.3.6  Integral Invariant    1.3.7  Applications to Dynamics of Ideal Fluid  1.4  Vakonomic Mechanics    1.4.1  Lagrange's Problem    1.4.2  Vakonomic Mechanics    1.4.3  Principle of Determinacy      1.4.4  Hamilton's Equations in Redundant Coordinates   1.5  Hamiltonian Formalism with Constraints      1.5.1  Dirac's Problem    1.5.2  Duality  '  1.6  Realization of Constraints    1.6.1  Various Methods of Realization of Constraints    1.6.2  Holonomic Constraints     1.6.3  Anisotropic Friction    1.6.4  Adjoint Masses    1.6.5  Adjoint Masses and Anisotropic Friction    1.6.6  Small Masses2 The n-Body Problem  2.1  The Two-Body Problem    2.1.1  Orbits    2.1.2  Anomalies    2.1.3  Collisions and Regularization    2.1.4  Geometry of Kepler's Problem  2.2  Collisions and Regularization    2.2.1  Necessary Condition for Stability    2.2.2  Simultaneous Collisions    2.2.3  Binary Collisions    2.2.4  Singularities of Solutions of the n-Body Problem  2.3  Particular Solutions    2.3.1  Central Configurations    2.3.2  Homographic Solutions    2.3.3  Effective Potential and Relative Equilibria    2.3.4  Periodic Solutions in the Case of Bodies cf Equal Masses  2.4  Final Motions in the Three-Body Problem    2.4.1  Classification of the Final Motions According to Chazy.    2.4.2  Symmetry of the Past and Future    2.5  Restricted Three-Body Problem    2.5.1  Equations of Motion. The Jacobi Integral    2.5.2  Relative Equilibria and Hill Regions    2.5.3  Hill's Problem  2.6  Ergodic Theorems of Celestial Mechanics    2.6.1  Stability in the Sense of Poisson    2.6.2  Probability of Capture  2.7  Dynamics in Spaces of Constant Curvature    2.7.1  Generalized Bertrand Problem    2.7.2  Kepler's Laws    2.7.3  Celestial Mechanics in Spaces of Constant Curvature    2.7.4  Potential Theory in Spaces of Constant Curvature3 Symmetry Groups and Order Reduction.  3.1  Symmetries and Linear Integrals    3.1.1  NSther's Theorem    3.1.2  Symmetries in Non-Holonomic Mechanics    3.1.3  Symmetries in Vakonomic Mechanics    3.1.4  Symmetries in Hamiltonian Mechanics  3.2  Reduction of Systems with Symmetries  ……4 Variational Principles and Methods5 Integrable Systems and Integration Methods6 Perturbation Theory for Integrable Systems7 Non-Integrable Systems8 Theory of Small Oscillations9 Tensor Invariants of Equations of DynamicsRecommended ReadingBibliographyIndex of NamesSubject Index

章節(jié)摘錄

插圖：This problem has many common features with the classical n-body prob-lem in Euclidean space. However, there are also essential differences. First,the two-body problem on S3 proves to be non-integrable: there are not suffi-ciently many first integrals for its solution and its orbits look quite complicated（see [137]）. Here the main difficulty is related to the fact that the Galileo-Newton law of inertia does not hold: the centre of mass of gravitating pointsno longer moves along an arc of a great circle.Furthermore, as in the classical case, binary collisions admit regularization.However, the question whether the generalized Sundman theorem is valid forthe three-body problem in spaces of constant curvature remains open. Thisquestion essentially reduces to the problem of elimination of triple collisions.Recall that in the ordinary three-body problem the absence of simultaneouscollisions is guaranteed by a non-zero constant value of the angular momentumof the system of n points with respect to their centre of mass （Theorem 2.3）.Of interest is the problem of finding partial solutions for n gravitatingbodies in spaces of constant curvature （similar to the classical solutions ofEuler and Lagrange）. Results in this direction can be found in the book [137].The restricted three-body problem was studied in this book: relative equilibriawere found and the Hill regions were constructed.

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