Geometric Mechanics: Toward a Unification of Classical Physics
Geometric Mechanics: Toward a Unification of Classical Physics
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几何力学：走向经典物理学的统一
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发表时间：
2007
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通讯作者：
R. Talman
中科院分区：
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作者：
R. Talman
前言。导言。参考书目。1经典力学与弦场理论综述。1.1预览和基本原理。1.2拉格朗日和哈密尔顿的回顾。1.3从哈密顿原理导出拉格朗日方程。1.4线性、多粒子系统。1.5等效势和开普勒问题。1.6多粒子系统。1.7串珠绳索的纵向摆动。1.8场理论处理和拉格朗日密度。1.9横向弦运动的哈密顿密度。1.10字符串运动表示为传播和反射波。1.11问题。参考书目。2力学几何，I，线性。2.1作为协变向量的平面对。2.2差分式。2.3代数张量。2.4度量几何中的笛卡尔向量(可能很复杂)。参考书目。3力学几何，II，曲线。3.1(实数)n维曲线坐标。3.2从绝对微分导出拉格朗日方程。3.3本征导数和双线性协变。3.4李导数-坐标方法。3.5李导数-李代数方法。3.6用微分算子识别向量场。3.7坐标同余。3.8 Lie-Drag同余和Lie导数。3.9拟基向量的交换子。参考书目。4力学几何，III，多线性。4.1广义欧几里得旋转和反射。4.2多向量。4.3欧几里得几何中的曲线坐标(续)。4.4三维空间中的旋量。5拉格朗日-庞加莱力学描述。5.1庞加莱方程。5.2庞加莱方程的变分推导。5.3用群论限制庞加莱方程。参考书目。6牛顿/规范不变力学。6.1向量力学。6.2规范不变形式的单粒子方程。6.3规范对刚体运动的不变描述。6.4福柯摆。6.5平底船和潜水员。参考书目。7几何光学的哈密顿处理。7.1力学和几何光学之间的类比。7.2变分原理。7.3傍轴光学、高斯光学、矩阵光学。7.4惠更斯原理。参考书目。8哈密顿-雅可比理论。8.1哈密顿-雅可比理论源于哈密顿原理。8.2使用哈密顿-雅各比方程确定弹道。8.3开普勒问题。8.4光学和量子力学之间的类比。参考书目。9相对论力学。9.1相对论运动学。9.2相对论力学。9.3将电磁力引入相对论力学。参考书目。10守恒定律和对称性。10.1线动量守恒。10.2角动量变化率：庞加莱方法。10.3角动量守恒：拉格朗日方法。10.4节约能源。10.5循环坐标和Routhian约化。10.6Noether‘s定理。10.7场论中的守恒定律。10.8从离散表示向连续表示过渡。10.9质点系统的角动量。10.10场的角动量。参考书目。11电磁理论。11.1电磁场张量。11.2电磁场方程。参考书目。12个相对论弦。12.1引言。12.2按公制表示的面积。12.3弦的拉格朗日密度和作用量。12.4运动方程、边界条件和非激发弦。12.5横向速度方面的动作。12.6按能量含量的正交参数化法。12.7自由开弦的一般运动。12.8一根旋转的直线。12.9保守的弦的Momenta。12.10光锥坐标。12.11相对论弦的振动模。参考书目。13.广义相对论。13.1引言。13.2局部惯性坐标的变换。13.3表面上的平行传输。13.4广义相对论中的孪生悖论。13.5曲率张量。13.6广义相对论的拉格朗日和能量动量张量。13.7爱因斯坦方程式的“推导”。13.8弱的非相对论引力。13.9Schwarzschild公制。13.10引力透镜和红移。参考书目。14个近似解析基。14.1规范变换。14.2与时间无关的正则变换。14.3动作-角度变量。14.4绝热不变性的例子。14.5绝热不变量守恒的准确性。14.6有条件的定期行动。参考书目。15个线性哈密顿系统。15.1线性哈密顿系统。15.2周期线性系统。参考书目。16微扰理论。16.1拉格朗日行星方程。16.2水星近日点的前移。16.3非简谐振动的迭代分析。16.4 Krylov和Bogoliubov的方法。16.5超收敛微扰理论。参考书目。17辛力学。17.1相空间的辛性。17.2辛几何。17.3标量函数的泊松括号。17.4积分不变量。17.5 Poincare-Cartan积分不变性I.17.6辛系演化。参考书目。索引。
Preface. Introduction. Bibliography. 1 Review of Classical Mechanics and String Field Theory. 1.1 Preview and Rationale. 1.2 Review of Lagrangians and Hamiltonians. 1.3 Derivation of the Lagrange Equation from Hamilton's Principle. 1.4 Linear, Multiparticle Systems. 1.5 Effective Potential and the Kepler Problem. 1.6 Multiparticle Systems. 1.7 Longitudinal Oscillation of a Beaded String. 1.8 Field Theoretical Treatment and Lagrangian Density. 1.9 Hamiltonian Density for Transverse String Motion. 1.10 String Motion Expressed as Propagating and ReflectingWaves. 1.11 Problems. Bibliography. 2 Geometry of Mechanics, I, Linear. 2.1 Pairs of Planes as Covariant Vectors. 2.2 Differential Forms. 2.3 Algebraic Tensors. 2.4 (Possibly Complex) Cartesian Vectors in Metric Geometry. Bibliography. 3 Geometry of Mechanics, II, Curvilinear. 3.1 (Real) Curvilinear Coordinates in n-Dimensions. 3.2 Derivation of the Lagrange Equations from the Absolute Differential. 3.3 Intrinsic Derivatives and the Bilinear Covariant. 3.4 The Lie Derivative - Coordinate Approach. 3.5 The Lie Derivative - Lie Algebraic Approach. 3.6 Identification of Vector Fields with Differential Operators. 3.7 Coordinate Congruences. 3.8 Lie-Dragged Congruences and the Lie Derivative. 3.9 Commutators of Quasi-Basis-Vectors. Bibliography. 4 Geometry of Mechanics, III, Multilinear. 4.1 Generalized Euclidean Rotations and Reflections. 4.2 Multivectors. 4.3 Curvilinear Coordinates in Euclidean Geometry (Continued). 4.4 Spinors in Three-Dimensional Space. 5 Lagrange-Poincare Description of Mechanics. 5.1 The Poincare Equation. 5.2 Variational Derivation of the Poincare Equation. 5.3 Restricting the Poincare Equation With Group Theory. Bibliography. 6 Newtonian/Gauge Invariant Mechanics. 6.1 Vector Mechanics. 6.2 Single Particle Equations in Gauge Invariant Form. 6.3 Gauge Invariant Description of Rigid Body Motion. 6.4 The Foucault Pendulum. 6.5 Tumblers and Divers. Bibliography. 7 Hamiltonian Treatment of Geometric Optics. 7.1 Analogy Between Mechanics and Geometric Optics. 7.2 Variational Principles. 7.3 Paraxial Optics, Gaussian Optics, Matrix Optics. 7.4 Huygens' Principle. Bibliography. 8 Hamilton-Jacobi Theory. 8.1 Hamilton-Jacobi Theory Derived from Hamilton's Principle. 8.2 Trajectory Determination Using the Hamilton-Jacobi Equation. 8.3 The Kepler Problem. 8.4 Analogies Between Optics and Quantum Mechanics. Bibliography. 9 Relativistic Mechanics. 9.1 Relativistic Kinematics. 9.2 Relativistic Mechanics. 9.3 Introduction of Electromagnetic Forces into Relativistic Mechanics. Bibliography. 10 Conservation Laws and Symmetry. 10.1 Conservation of Linear Momentum. 10.2 Rate of Change of Angular Momentum: Poincare Approach. 10.3 Conservation of Angular Momentum: Lagrangian Approach. 10.4 Conservation of Energy. 10.5 Cyclic Coordinates and Routhian Reduction. 10.6 Noether's Theorem. 10.7 Conservation Laws in Field Theory. 10.8 Transition From Discrete to Continuous Representation. 10.9 Angular Momentum of a System of Particles. 10.10 Angular Momentum of a Field. Bibliography. 11 Electromagnetic Theory. 11.1 The Electromagnetic Field Tensor. 11.2 The Electromagnetic Field Equations. Bibliography. 12 Relativistic Strings. 12.1 Introduction. 12.2 Area Representation in Terms of the Metric. 12.3 The Lagrangian Density and Action for Strings. 12.4 Equations of Motion, Boundary Conditions, and Unexcited Strings. 12.5 The Action in Terms of Transverse Velocity. 12.6 Orthogonal Parameterization by Energy Content. 12.7 General Motion of a Free Open String. 12.8 A Rotating Straight String. 12.9 Conserved Momenta of a String. 12.10 Light Cone Coordinates. 12.11 Oscillation Modes of a Relativistic String. Bibliography. 13 General Relativity. 13.1 Introduction. 13.2 Transformation to Locally Inertial Coordinates. 13.3 Parallel Transport on a Surface. 13.4 The Twin Paradox in General Relativity. 13.5 The Curvature Tensor. 13.6 The Lagrangian of General Relativity and the Energy-Momentum Tensor. 13.7 "Derivation" of the Einstein Equation. 13.8 Weak, Nonrelativistic Gravity. 13.9 The Schwarzschild Metric. 13.10 Gravitational Lensing and Red Shifts. Bibliography. 14 Analytic Bases for Approximation. 14.1 Canonical Transformations. 14.2 Time-Independent Canonical Transformation. 14.3 Action-Angle Variables. 14.4 Examples of Adiabatic Invariance. 14.5 Accuracy of Conservation of Adiabatic Invariants. 14.6 Conditionally Periodic Motion. Bibliography. 15 Linear Hamiltonian Systems. 15.1 Linear Hamiltonian Systems. 15.2 Periodic Linear Systems. Bibliography. 16 Perturbation Theory. 16.1 The Lagrange Planetary Equations. 16.2 Advance of Perihelion of Mercury. 16.3 Iterative Analysis of Anharmonic Oscillations. 16.4 The Method of Krylov and Bogoliubov. 16.5 Superconvergent Perturbation Theory. Bibliography. 17 Symplectic Mechanics. 17.1 The Symplectic Properties of Phase Space. 17.2 Symplectic Geometry. 17.3 Poisson Brackets of Scalar Functions. 17.4 Integral Invariants. 17.5 Invariance of the Poincare-Cartan Integral Invariant I.I. 17.6 Symplectic System Evolution. Bibliography. Index.