Mathematical Aspects of Classical and Celestial Mechanics

Basic Principles of Classical Mechanics     1Newtonian Mechanics     1Space, Time, Motion     1Newton-Laplace Principle of Determinacy     2Principle of Relativity     9Principle of Relativity and Forces of Inertia     12Basic Dynamical Quantities. Conservation Laws     15Lagrangian Mechanics     17Preliminary Remarks     17Variations and Extremals     19Lagrange's Equations     21Poincare's Equations     23Motion with Constraints     26Hamiltonian Mechanics     30Symplectic Structures and Hamilton's Equations     30Generating Functions     33Symplectic Structure of the Cotangent Bundle     34The Problem of n Point Vortices     35Action in the Phase Space     37Integral Invariant     38Applications to Dynamics of Ideal Fluid     40Vakonomic Mechanics     41Lagrange's Problem     42Vakonomic Mechanics     43Principle of Determinacy     46Hamilton's Equations in Redundant Coordinates     47Hamiltonian Formalism with Constraints     48Dirac's Problem     48Duality     50Realization of Constraints     51Various Methods of Realization of Constraints     51Holonomic Constraints     52Anisotropic Friction     54Adjoint Masses     55Adjoint Masses and Anisotropic Friction     58Small Masses     59The n-Body Problem     61The Two-Body Problem     61Orbits     61Anomalies     67Collisions and Regularization     69Geometry of Kepler's Problem     71Collisions and Regularization     72Necessary Condition for Stability     72Simultaneous Collisions     73Binary Collisions     74Singularities of Solutions of the n-Body Problem     78Particular Solutions     79Central Configurations     79Homographic Solutions     80Effective Potential and Relative Equilibria     82Periodic Solutions in the Case of Bodies of Equal Masses     82Final Motions in the Three-Body Problem     83Classification of the Final Motions According to Chazy     83Symmetry of the Past and Future     84Restricted Three-Body Problem     86Equations of Motion. The Jacobi Integral     86Relative Equilibria and Hill Regions     87Hill's Problem     88Ergodic Theorems of Celestial Mechanics     92Stability in the Sense of Poisson     92Probability of Capture     94Dynamics in Spaces of Constant Curvature     95Generalized Bertrand Problem     95Kepler's Laws     96Celestial Mechanics in Spaces of Constant Curvature     97Potential Theory in Spaces of Constant Curvature     98Symmetry Groups and Order Reduction     103Symmetries and Linear Integrals     103Nother's Theorem     103Symmetries in Non-Holonomic Mechanics     107Symmetries in Vakonomic Mechanics     109Symmetries in Hamiltonian Mechanics     110Reduction of Systems with Symmetries     111Order Reduction (Lagrangian Aspect)     111Order Reduction (Hamiltonian Aspect)     116Examples: Free Rotation of a Rigid Body and the Three-Body Problem     122Relative Equilibria and Bifurcation of Integral Manifolds     126Relative Equilibria and Effective Potential     126Integral Manifolds, Regions of Possible Motion, and Bifurcation Sets     128The Bifurcation Set in the Planar Three-Body Problem     130Bifurcation Sets and Integral Manifolds in the Problem of Rotation of a Heavy Rigid Body with a Fixed Point     131Variational Principles and Methods     135Geometry of Regions of Possible Motion     136Principle of Stationary Abbreviated Action     136Geometry of a Neighbourhood of the Boundary     139Riemannian Geometry of Regions of Possible Motion with Boundary     140Periodic Trajectories of Natural Mechanical Systems     145Rotations and Librations     145Librations in Non-Simply-Connected Regions of Possible Motion     147Librations in Simply Connected Domains and Seifert's Conjecture     150Periodic Oscillations of a Multi-Link Pendulum     153Periodic Trajectories of Non-Reversible Systems     156Systems with Gyroscopic Forces and Multivalued Functionals     156Applications of the Generalized Poincare Geometric Theorem     159Asymptotic Solutions. Application to the Theory of Stability of Motion     161Existence of Asymptotic Motions     162Action Function in a Neighbourhood of an Unstable Equilibrium Position     165Instability Theorem     166Multi-Link Pendulum with Oscillating Point of Suspension     167Homoclinic Motions Close to Chains of Homoclinic Motions     168Integrable Systems and Integration Methods     171Brief Survey of Various Approaches to Integrability of Hamiltonian Systems     171Quadratures     171Complete Integrability     174Normal Forms     176Completely Integrable Systems     179Action-Angle Variables     179Non-Commutative Sets of Integrals     183Examples of Completely Integrable Systems     185Some Methods of Integration of Hamiltonian Systems     191Method of Separation of Variables     191Method of L-A Pairs     197Integrable Non-Holonomic Systems     199Differential Equations with Invariant Measure     199Some Solved Problems of Non-Holonomic Mechanics     202Perturbation Theory for Integrable Systems     207Averaging of Perturbations     207Averaging Principle     207Procedure for Eliminating Fast Variables. Non-Resonant Case     211Procedure for Eliminating Fast Variables. Resonant Case      216Averaging in Single-Frequency Systems     217Averaging in Systems with Constant Frequencies     226Averaging in Non-Resonant Domains     229Effect of a Single Resonance     229Averaging in Two-Frequency Systems     237Averaging in Multi-Frequency Systems     242Averaging at Separatrix Crossing     244Averaging in Hamiltonian Systems     256Application of the Averaging Principle     256Procedures for Eliminating Fast Variables     265KAM Theory     273Unperturbed Motion. Non-Degeneracy Conditions     273Invariant Tori of the Perturbed System     274Systems with Two Degrees of Freedom     279Diffusion of Slow Variables in Multidimensional Systems and its Exponential Estimate     286Diffusion without Exponentially Small Effects     292Variants of the Theorem on Invariant Tori     294KAM Theory for Lower-Dimensional Tori     297Variational Principle for Invariant Tori. Cantori     307Applications of KAM Theory     311Adiabatic Invariants     314Adiabatic Invariance of the Action Variable in Single-Frequency Systems     314Adiabatic Invariants of Multi-Frequency Hamiltonian Systems     323Adiabatic Phases     326Procedure for Eliminating Fast Variables. Conservation Time of Adiabatic Invariants     332Accuracy of Conservation of Adiabatic Invariants     334Perpetual Conservation of Adiabatic Invariants     340Adiabatic Invariants in Systems with Separatrix Crossings     342Non-Integrable Systems     351Nearly Integrable Hamiltonian Systems     351The Poincare Method     352Birth of Isolated Periodic Solutions as an Obstruction to Integrability     354Applications of Poincare's Method     358Splitting of Asymptotic Surfaces     360Splitting Conditions. The Poincare Integral     360Splitting of Asymptotic Surfaces as an Obstruction to Integrability     366Some Applications     370Quasi-Random Oscillations     373Poincare Return Map     375Symbolic Dynamics     378Absence of Analytic Integrals     380Non-Integrability in a Neighbourhood of an Equilibrium Position (Siegel's Method)     381Branching of Solutions and Absence of Single-Valued Integrals     385Branching of Solutions as Obstruction to Integrability      385Monodromy Groups of Hamiltonian Systems with Single-Valued Integrals     388Topological and Geometrical Obstructions to Complete Integrability of Natural Systems     391Topology of Configuration Spaces of Integrable Systems     392Geometrical Obstructions to Integrability     394Multidimensional Case     396Ergodic Properties of Dynamical Systems with Multivalued Hamiltonians     396Theory of Small Oscillations     401Linearization     401Normal Forms of Linear Oscillations     402Normal Form of a Linear Natural Lagrangian System     402Rayleigh-Fisher-Courant Theorems on the Behaviour of Characteristic Frequencies when Rigidity Increases or Constraints are Imposed     403Normal Forms of Quadratic Hamiltonians     404Normal Forms of Hamiltonian Systems near an Equilibrium Position     406Reduction to Normal Form     406Phase Portraits of Systems with Two Degrees of Freedom in a Neighbourhood of an Equilibrium Position at a Resonance     409Stability of Equilibria of Hamiltonian Systems with Two Degrees of Freedom at Resonances     416Normal Forms of Hamiltonian Systems near Closed Trajectories     417Reduction to Equilibrium of a System with Periodic Coefficients     417Reduction of a System with Periodic Coefficients to Normal Form     418Phase Portraits of Systems with Two Degrees of Freedom near a Closed Trajectory at a Resonance     419Stability of Equilibria in Conservative Fields     422Lagrange-Dirichlet Theorem     422Influence of Dissipative Forces     426Influence of Gyroscopic Forces     427Tensor Invariants of Equations of Dynamics     431Tensor Invariants     431Frozen-in Direction Fields     431Integral Invariants     433Poincare-Cartan Integral Invariant     436Invariant Volume Forms     438Liouville's Equation     438Condition for the Existence of an Invariant Measure     439Application of the Method of Small Parameter     442Tensor Invariants and the Problem of Small Denominators     445Absence of New Linear Integral Invariants and Frozen-in Direction Fields     445Application to Hamiltonian Systems     446Application to Stationary Flows of a Viscous Fluid     449Systems on Three-Dimensional Manifolds     451Integral Invariants of the Second Order and Multivalued Integrals     455Tensor Invariants of Quasi-Homogeneous Systems      457Kovalevskaya-Lyapunov Method     457Conditions for the Existence of Tensor Invariants     459General Vortex Theory     461Lamb's Equation     461Multidimensional Hydrodynamics     463Invariant Volume Forms for Lamb's Equations     465Recommended Reading     471Bibliography     475Index of Names     507Subject Index     511