动态系统与流体流的几何理论GEOMETRICAL THEORY OF DYNAMICAL SYSTEMS AND FLUID FLOWS

2004-12  

World Scientific Publishing Company (2004年11月1日)  

Tsutomu Kambe  

416  

This is an introductory textbook on the geometrical theory of dynamical systems, fluid flows, and certain integrable systems. The subjects are interdisciplinary and extend from mathematics, mechanics and physics to mechanical engineering, and the approach is very fundamental. The underlying concepts are based on differential geometry and theory of Lie groups in the mathematical aspect, and on transformation symmetries and gauge theory in the physical aspect. A great deal of effort has been directed toward making the description elementary, clear and concise, so that beginners will have an access to the topics.

PrefaceⅠ Mathematical Bases　1. Manifolds, Flows, Lie Groups and Lie Algebras　　1.1 Dynamical Systems 1.2 Manifolds and Diffeomorphisms 1.3 Flows and Vector Fields 　1.3.1 A steady flow and its velocity field 1.3.2 Tangent vector and differential operator 1.3.3 Tangent space 1.3.4 Time-dependent (unsteady) velocity field　　1.4 Dynamical Trajectory 1.4.1 Fiber bundle (tangent bundle) 1.4.2 Lagrangian and Halniltonian 1.4.3 Legendre transformation 1.5 Differential and Inner Product 1.5.1 Covector (1-form) 1.5.2 Inner (scalar) product 1.6 Mapping of Vectors and Covectors 1.6.1 Push-forward transformation 1.6.2 Pull-back transformation 1.6.3 Coordinate transformation 1.7 Lie Group and Invariant Vector Fields 1.8 Lie Algebra and Lie Derivative 1.8.1 Lie algebra, adjoint operator and Lie bracket 1.8.2 An example of the rotation group SO(3) 1.8.3 Lie derivative and Lagrange derivative 1.9 Diffeomorphisms of a Circle S1 1.10 Transformation of Tensors and Invariance 1.10.1 Transformations of vectors and metric tensors . . . 1.10.2 Covariant tensors 1.10.3 Mixed tensors 1.10.4 Contravariant tensors　2. Geometry of Surfaces in R3 2.1 First Fundamental Form 2.2 Second Fundamental Form 2.3 Gauss's Surface Equation and an Induced Connection 2.4 Gauss Mainardi Codazzi Equation and Integrability . . 2.5 Gaussian Curvature of a Surface 2.5.1 Riemann tensors 2.5.2 Gaussian curvature 2.5.3 Geodesic curvature and normal curvature 2.5.4 Principal curvatures 2.6 Geodesic Equation 2.7 Structure Equations in Differential Forms 2.7.1 Smooth surfaces in IRa and integrability 2.7.2 Structure equations 2.7.3 Geodesic equation 2.8 Gauss Spherical Map 2.9 Gauss Bonnet Theorem I 2.10 Gauss Bonnet Theorem II 2.11 Uniqueness: First and Second Fundamental Tensors 3. Riemannian Geometry 3.1 Tangent Space 3.1.1 Tangent vectors and inner product 3.1.2 Riemannian metric 3.1.3 Examples of metric tensor　……Ⅱ Dynamical SystemsⅢ　Flows of Ideal FluidsⅣ　Geometry of Integrable SystemsAppendix A Topological Space and MappingsAppendix B Exterior Forms, Products and DifferentialsAppendix C Lie Groups and Rotation GroupsAppendix D A Curve and a Surface in R3Appendix E Curvature TransformationAppendix F Function Spaces Lp, Hs and Orthogonal DecompositionAppendix G Derivation of KdV Equation of a Shallow Water WaveAppendix H Two-Cocycle, Central Extension and Bott CocycleAppendix I Additional Comment on the Gauge THeory of 7.3Appendix J Frobenius Integration Theorem and Pfaffian SystemAppendix K Orthogonal Coordinate Net and Lines of CurvatureReferencesIndex

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