力學(xué)

前言

　　Purpose and Emphasis. Mechanics not only is the oldest branch of physics but was and still is the basis for all of theoretical physics. Quantum mechanics can hardly be understood, perhaps cannot even be formulated, without a good knowl- edge of general mechanics. Field theories such as electrodynamics borrow their formal framework and many of their building principles from mechanics. In short, throughout the many modern developments of physics where one frequently turns back to the principles of classical mechanics its model character is felt. For this reason it is not surprising that the presentation of mechanics reflects to some ex- tent the development of modern physics and that today this classical branch of theoretical physics is taught rather differently than at the time of Arnold Som- merfeld, in the 1920s, or even in the 1950s, when more emphasis was put on the theory and the applications of partial-differential equations. Today, symmetries and invariance principles, the structure of the space-time continuum, and the geomet- rical structure of mechanics play an important role. The beginner should realize that mechanics is not primarily the art of describing block-and-tackles, collisions of billiard balls, constrained motions of the cylinder in a washing machine, or bi- cycle riding. However fascinating such systems may be, mechanics is primarily the field where one learns to develop general principles from which equations of motion may be derived, to understand the importance of symmetries for the dy- namics, and, last but not least, to get some practice in using theoretical tools and concepts that are essential for all branches of physics.　　Besides its role as a basis for much of theoretical physics and as a training ground for physical concepts, mechanics is a fascinating field in itself. It is not easy to master, for the beginner, because it has many different facets and its structure is less homogeneous than, say, that of electrodynamics. On a first assault one usually does not fully realize both its charm and its difficulty. Indeed, on returning to various aspects of mechanics, in the course of one's studies, one will be surprised to discover again and again that it has new facets and new secrets. And finally, one should be aware of the fact that mechanics is not a closed subject, lost forever in the archives of the nineteenth century. As the reader will realize in Chap. 6, if he or she has not realized it already, mechanics is an exciting field of research with many important questions of qualitative dynamics remaining unanswered.

內(nèi)容概要

　　Purpose and Emphasis. Mechanics not only is the oldest branch of physics but was and still is the basis for all of theoretical physics. Quantum mechanics can hardly be understood, perhaps cannot even be formulated, without a good knowl- edge of general mechanics.

書籍目錄

1. Elementary Newtonian Mechanics1.1 Newton's Laws （1687） and Their Interpretation1.2 Uniform Rectilinear Motion and Inertial Systems1.3 Inertial Frames in Relative Motion!.4 Momentum and Force1.5 Typical Forces. A Remark About Units1.6 Space, Time, and Forces1.7 The Two-Body System with Internal Forces1.7.1 Center-of-Mass and Relative Motion1.7.2 Example: The Gravitational Force Between Two Celestial Bodies （Kepler's Problem）1.7.3 Center-of-Mass and Relative Momentum in the Two-Body System1.8 Systems of Finitely Many Particles1.9 The Principle of Center-of-Mass Motion1.10 The Principle of Angular-Momentum Conservation1.11 The Principle of Energy Conservation1.12 The Closed n-Particle System1.13 Galilei Transformations1.14 Space and Time with Galilei Invariance1.15 Conservative Force Fields1.16 One-Dimensional Motion of a Point Particle1.17 Examples of Motion in One Dimension1.17.1 The Harmonic Oscillator1.17.2 The Planar Mathematical Pendulum1.18 Phase Space for the n-Particle System （in R3）1.19 Existence and Uniqueness of the Solutions of x=F（x, t）1.20 Physical Consequences of the Existence and Uniqueness Theorem1.21 Linear Systems1.21.1 Linear, Homogeneous Systems1.21.2 Linear, Inhomogeneous Systems1.22 Integrating One-Dimensional Equations of Motion1.23 Example: The Planar Pendulum for Arbitrary Deviations from the Vertical1.24 Example: The Two-Body System with a Central Force1.25 Rotating Reference Systems: Coriolis and Centrifugal Forces1.26 Examples of Rotating Reference Systems1.27 Scattering of Two Particles that Interact via a Central Force Kinematics1.28 Two-Particle Scattering with a Central Force: Dynamics1.29 Example: Coulomb Scattering of Two Particles with Equal Mass and Charge1.30 Mechanical Bodies of Finite Extension1.31 Time Averages and the Virial TheoremAppendix: Practical Examples2. The Principles of Canonieal Mechanics2.1 Constraints and Generalized Coordinates2.1.1 Definition of Constraints2.1.2 Generalized Coordinates2.2 D'Alembert's Principle2.2.1 Definition of Virtual Displacements2.2.2 The Static Case2.2.3 The Dynamical Case2.3 Lagrange's Equations2.4 Examples of the Use of Lagrange's Equations2.5 A Digression on Variational Principles2.6 Hamilton's Variational Principle （1834）2.7 The Euler-Lagrange Equations2.8 Further Examples of the Use of Lagrange's Equations2.9 A Remark About Nonuniqueness of the Lagrangian Function .2.10 Gauge Transformations of the Lagrangian Function2.11 Admissible Transformations of the Generalized Coordinates2.12 The Hamiltonian Function and Its Relation to the Lagrangian Function L2.13 The Legendre Transformation for the Case of One Variable2.14 The Legendre Transformation for the Case of Several Variables2.15 Canonical Systems2.16 Examples of Canonical Systems2.17 The Variational Principle Applied to the Hamiltonian Function2.18 Symmetries and Conservation Laws2.19 Noether's Theorem2.20 The Generator for Infinitesimal Rotations About an Axis2.21 More About the Rotation Group2.22 Infinitesimal Rotations and Their Generators2.23 Canonical Transformations2.24 Examples of Canonical Transformations2.25 The Structure of the Canonical Equations2.26 Example: Linear Autonomous Systems in One Dimension2.27 Canonical Transformations in Compact Notation2.28 On the Symplectic Structure of Phase Space2.29 Liouville's Theorem2.29.1 The Local Form2.29.2 The Global Form2.30 Examples for the Use of Liouviile's Theorem2.31 Poisson Brackets2.32 Properties of Poisson Brackets2.33 Infinitesimal Canonical Transformations2.34 Integrals of the Motion2.35 The Hamilton-Jacobi Differential Equation2.36 Examples for the Use of the Hamilton-Jacobi Equation2.37 The Hamilton-Jacobi Equation and Integrable Systems2.37.1 Local Rectification of Hamiltonian Systems2.37.2 Integrable Systems2.37.3 Angle and Action Variables2.38 Perturbing Quasiperiodic Hamiltonian Systems2.39 Autonomous, Nondegenerate Hamiltonian Systems in the Neighborhood of Integrable Systems2.40 Examples. The Averaging Principle2.40.1 The Anharmonic Oscillator2.40.2 Averaging of Perturbations2.41 Generalized Theorem of NoetherAppendix: Practical Examples3. The Mechanics of Rigid Bodies3.1 Definition of Rigid Body3.2 Infinitesimal Displacement of a Rigid Body3.3 Kinetic Energy and the Inertia Tensor3.4 Properties of the Inertia Tensor3.5 Steiner's Theorem3.6 Examples of the Use of Steiner's Theorem3.7 Angular Momentum of a Rigid Body3.8 Force-Free Motion of Rigid Bodies3.9 Another Parametrization of Rotations: The Euler Angles3.10 Definition of Eulerian Angles3.11 Equations of Motion of Rigid Bodies3.12 Euler's Equations of Motion3.13 Euler's Equations Applied to a Force-Free Top3.14 The Motion of a Free Top and Geometric Constructions3.15 The Rigid Body in the Framework of Canonical Mechanics3.16 Example: The Symmetric Children's Top in a Gravitational Field3.17 More About the Spinning Top3.18 Spherical Top with Friction: The "Tippe Top"3.18.1 Conservation Law and Energy Considerations3.18.2 Equations of Motion and Solutions with Constant EnergyAppendix：PracticaI Examples4. Relativistic Mechanics4.1 Failures of Nonrelativistic Mechanics4.2 Constancy of the Speed of Light4.3 The Lorentz Transformations4.4 Analysis or Lorentz and Poincar6 Transformations4.4.1 Rotations and Specia!Lorentz Tranformations （“Boosts”）4.4.2 Interpretation of Specia!Lorentz Transformations4.5 Decomposition 0f Lorentz Transformations into 7heir Components4.5.1 Proposition on Orthochronous Proper Lorentz Transformations4.5.2 Corollary of the Decomposition Theorem and Some Consequences4.6 Addition of Relativistic VeIocities4.7 Galilean and Lorentzian Space-Time ManifoIds4.8 Orbita!Curves and Proper Time4.9 Relativistic Dynamics4.9.1 Newton’S Equation4.9.2 The Energy-Momentum Vector4.9.3 The Lorentz Force4.10 Time Dilatation and Scale Contraction4.11 More About the Motion of Free Particles4.12 The Conformal Group5. Geometric Aspects of Mechanics5.1 Manifoids of Generalized COOrdinates5.2 Differentiable ManifoIds5.2.1 The Euclidean Space R5.2.2 Smooth or Differentiable Manifoids5.2.3 Examples of Smooth ManifoIds5.3 GeometricalObiects on ManifoIds5.3.1 Functions and Curves on ManifoIds5.3.2 Tangent Vectors on a Smooth ManifoId5.3.3 The Tangent Bundle of a Manifoid5.3.4 Vector Fields on Smooth ManifoIds5.3.5 Exterior Forms5.4 Caiculus on ManifoIds5.4.1 Differentiable Mappings of ManifoIds5.4.2 Integra!Curves of Vector FieldsStability and ChaodExercisesSolution of ExercisesAuthor IndexSubject Index

章節(jié)摘錄

　　By assumption the transformation matrix is not singular; cf. （2.34）. This proves the proposition.　　Another way of stating this result is this: the variational derivatives are covariant under diffeomorphic transformations of the generalized coordinates.　　It is not correct, therefore, to state that the Lagrangian function is "T - U". Although this is a natural form, in those cases where kinetic and potential energies are defined it is certainly not the only one that describes a given problem. In gen- eral, L is a function of q and q', as well as of time t, and no more. How to construct a Lagrangian function is more a question of the symmetries and invariances of the physical system one wishes to describe. There may well be cases where there is no kinetic energy or no potential energy, in the usual sense, but where a Lagrangian can be found, up to gauge transformations （2.33）, which gives the correct equa- tions of motion. This is true, in particular, in applying the variational principle of Hamilton to theories in which fields take over the role of dynamical variables. For such theories, the notion of kinetic and potential parts in the Lagrangian must be generalized anyway, if they are defined at all.　　The proposition proved above tells us that with any set of generalized coordi- nates there is an infinity of other, equivalent sets of variables. Which set is chosen in practice depends on the peculiarities of the system under consideration. For ex- ample, a clever choice will be one where as many integrals of the motion as possi- ble will be manifest. We shall say more about this as well as about the geometric meaning of this multiplicity later. For the moment we note that the transforma- tions must be diffeomorphisms. In transforming to new coordinates we wish to conserve the number of degrees of freedom as well as the differential structure of the system. Only then can the physics be independent of the special choice of variables.　　……

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