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.
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目录
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 Poincaré'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 Masses 2 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 of 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 Curvature 3 Symmetry Groups and Order Reduction 3.1 Symmetries and Linear Integrals 3.1.1 Nöther'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 3.2.1 Order Reduction (Lagrangian Aspect) 3.2.2 Order Reduction (Hamiltonian Aspect) 3.2.3 Examples: Free Rotation of a Rigid Body and the Three-Body Problem 3.3 Relative Equilibria and Bifurcation of Integral Manifolds 3.3.1 Relative Equilibria and Effective Potential 3.3.2 Integral Manifolds，Regions of Possible Motion，and Bifurcation Sets 3.3.3 The Bifurcation Set in the Planar Three-Body Problem 3.3.4 Bifurcation Sets and Integral Manifolds in the Problem of Rotation of a Heavy Rigid Body with a Fixed Point 4 Variational Principles and Methods 4.1 Geometry of Regions of Possible Motion 4.1.1 Principle of Stationary Abbreviated Action 4.1.2 Geometry of a Neighbourhood of the Boundary 4.1.3 Riemannian Geometry of Regions of Possible Motion with Boundary 4.2 Periodic Trajectories of Natural Mechanical Systems 4.2.1 Rotations and Librations 4.2.2 Librations in Non-Simply-Connected Regions of Possible Motion 4.2.3 Librations in Simply Connected Domains and Seifert's Conjecture 4.2.4 Periodic Oscillations of a Multi-Link Pendulum 4.3 Periodic Trajectories of Non-Reversible Systems 4.3.1 Systems with Gyroscopic Forces and Multivalued Functionals 4.3.2 Applications of the Generalized Poincaré Geometric Theorem 4.4 Asymptotic Solutions. Application to the Theory of Stability of Motion 4.4.1 Existence of Asymptotic Motions 4.4.2 Action Function in a Neighbourhood of an Unstable Equilibrium Position 4.4.3 Instability Theorem 4.4.4 Multi-Link Pendulum with Oscillating Point of Suspension 4.4.5 Homoclinic Motions Close to Chains of Homoclinic Motions 5 Integrable Systems and Integration Methods 5.1 Brief Survey of Various Approaches to Integrability of Hamiltonian Systems 5.1.1 Quadratures 5.1.2 Complete Integrability 5.1.3 Normal Forms 5.2 Completely Integrable Systems 5.2.1 Action-Angle Variables 5.2.2 Non-Commutative Sets of Integrals 5.2.3 Examples of Completely Integrable Systems 5.3 Some Methods of Integration of Hamiltonian Systems 5.3.1 Method of Separation of Variables 5.3.2 Method of L-A Pairs 5.4 Integrable Non-Holonomic Systems 5.4.1 Differential Equations with Invariant Measure 5.4.2 Some Solved Problems of Non-Holonomic Mechanics 6 Perturbation Theory for Integrable Systems 6.1 Averaging of Perturbations 6.1.1 Averaging Principle 6.1.2 Procedure for Eliminating Fast Variables. Non-Resonant Case 6.1.3 Procedure for Eliminating Fast Variables. Resonant Case 6.1.4 Averaging in Single-Frequency Systems 6.1.5 Averaging in Systems with Constant Frequencies 6.1.6 Averaging in Non-Resonant Domains 6.1.7 Effect of a Single Resonance 6.1.8 Averaging in Two-Frequency Systems 6.1.9 Averaging in Multi-Frequency Systems 6.1.10 Averaging at Separatrix Crossing 6.2 Averaging in Hamiltonian Systems 6.2.1 Application of the Averaging Principle 6.2.2 Procedures for Eliminating Fast Variables 6.3 KAM Theory 6.3.1 Unperturbed Motion . Non-Degeneracy Conditions 6.3.2 Invariant Tori of the Perturbed System 6.3.3 Systems with Two Degrees of Freedom 6.3.4 Diffusion of Slow Variables in Multidimensional Systems and its Exponential Estimate 6.3.5 Diffusion without Exponentially Small Effects 6.3.6 Variants of the Theorem on Invariant Tori 6.3.7 KAM Theory for Lower-Dimensional Tori 6.3.8 Variational Principle for Invariant Tori. Cantori 6.3.9 Applications of KAM Theory 6.4 Adiabatic Invariants 6.4.1 Adiabatic Invariance of the Action Variable in Single-Frequency Systems 6.4.2 Adiabatic Invariants of Multi-Frequency Hamiltonian Systems 6.4.3 Adiabatic Phases 6.4.4 Procedure for Eliminating Fast Variables. Conservation Time of Adiabatic Invariants 6.4.5 Accuracy of Conservation of Adiabatic Invariants 6.4.6 Perpetual Conservation of Adiabatic Invariants 6.4.7 Adiabatic Invariants in Systems with Separatrix Crossings 7 Non-Integrable Systems 7.1 Nearly Integrable Hamiltonian Systems 7.1.1 The Poincare Method 7.1.2 Birth of Isolated Periodic Solutions as an Obstruction to Integrability 7.1.3 Applications of Poincare's hlethod 7.2 Splitting of Asymptotic Surfaces 7.2.1 Splitting Conditions. The Poincare Integral 7.2.2 Splitting of Asymptotic Surfaces as an Obstruction to Integrability 7.2.3 Some Applications 7.3 Quasi-Random Oscillations 7.3.1 Poincaré Return Map 7.3.2 Symbolic Dynamics 7.3.3 Absence of Analytic Integrals 7.4 Non-Integrability in a Neighbourhood of an Equilibrium Position (Siegel's Method) 7.5 Branching of Solut. ions and Absence of Single-Valued Integrals 7.5.1 Branching of Solutions as Obstruction to Integrability 7.5.2 Monodromy Groups of Hamiltonian Systems with Single-Valued Integrals 7.6 Topological and Geometrical Obstructions to Complete Integrability of Natural Systems 7.6.1 Topology of Configuration Spaces of Integrable Systems 7.6.2 Geometrical Obstructions to Integrability 7.6.3 hlultidimensional Case 7.6.4 Ergodic Properties of Dynamical Systems with hlultivalued Hamiltonians 8 Theory of Small Oscillations 8.1 Linearization 8.2 Normal Forms of Linear Oscillations 8.2.1 Normal Form of a Linear Natural Lagrangian System 8.2.2 Rayleigh-Fisher-Courant Theorems on the Behaviour of Characteristic Frequencies when Rigidity Increases or Constraints are Irriposetl 8.2.3 Normal Forms of Quadratic Hamiltonians 8.3 Nornial Forms of Hamiltonian Systems near an Equilibrium Position 8.3.1 Reduct. ion to Normal Form 8.3.2 Phase Portraits of Systenis with Two Degrees of Freedonl in a Neighbolirhood of an Eqliilibriuim Position at a R.esonance 8.3.3 Stability of Equilibria of Hamiltonian Systenis with Two Degrees of Freedom at Resonances 8.4 Normal Forms of Harniltonian Systems near Closed Trajectories 8.4.1 Reduction to Equilibriurn of a System with Periodic Coefficients 8.4.2 Reduction of a System with Periodic Coefficients to NormalForm 8.4.3 Phase Portraits of Systems with Two Degrees of Freedom near a Closed Trajectory at a Resonance 8.5 Stability of Equilibria in Conservative Fields 8.5.1 Lagrange-Dirichlet Theorem 8.5.2 Influence of Dissipative Forces 8.5.3 Influence of Gyroscopic Forces 9 Tensor Invariants of Equations of Dynamics 9.1 Tensorlnvariants 9.1.1 Frozen-in Direction Fields 9.1.2 Integral Invariants 9.1.3 Poincaré-Cartan Integral Invariant 9.2 Invariant Volume Fornls 9.2.1 Liouville's Equation 9.2.2 Condition for the Existence of an Invariant Measure 9.2.3 Application of the Method of Small Parametcr 9.3 Tensor Invariants and the Problem of Small Denominators 9.3.1 Absence of New Linear Integral Invariants and Frozen-in Direction Fields 9.3.2 Application to Hamiltonian Systems 9.3.3 Application to Stationary Flows of a Viscous Fluid 9.4 Systerrls on Three-Dimensional Manifolds 9.5 Integral Invariants of the Second Order:md Multivalued Integrals 9.6 Tensor Invariants of Quasi-Homogeneous Systems 9.6.1 Kovalevskaya-Lyapunov Method 9.6.2 Condit. ions for the Existence of Tensor Invariants 9.7 General Vortex Theory 9.7.1 Lamb's Equation 9.7.2 hlultidimensional Hydrodynamics 9.7.3 Invariant Volume Forms for Lamb's Equations Recommended Reading Bibliography Index of Names Subject Index
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