经典力学和天体力学中的数学论题

Chapter 1. Basic Principles of Classical Mechanics
1. Newtonian Mechanics
1.1. Space. Time, Motion
1.2. The Newton-Laplace Principle of Determinacy
1.3. The Principle of Relativity
1.4. Basic Dynamical Quantities. Conservation Laws
2. Lagrangian Mechanics
2.1. Preliminary Remarks
2.2. Variations and Extremals
2.3. Lagrange''s Equations
2.4. Poincare''s Equations
2.5. Constrained Motion
3. Hamiltonian Mechanics
3.1. Symplectic Structures and Hamilton''s Equations
3.2. Generating Functions
3.3. Symplectic Structure of the Cotangent Bundle
3.4. The Problem of n Point Vortices
3.5. The Action Functional in Phase Space
3.6. Integral Invariants
3.7. Applications to the Dynamics of Ideal Fluids
3.8. Principle of Stationary Isoenergetic Action
4. Vakonomic Mechanics
4.1. Lagrange''s Problem
4.2. Vakonomic Mechanics
4.3. The Principle of Determinacy
4.4. Hamilton''s Equations in Redundant Coordinates
5. Hamiltonian Formalism with Constraints
5.1. Dirac''s Problem
5.2. Duality
6. Realization of Constraints
6.1. Various Methods of Realizing Constraints
6.2. Holonomic Constraints
6.3. Anisotropic Friction
6.4. Adjoining Masses
6.5. Adjoining Masses and Anisotropic Friction
6.6. Small Masses
Chapter 2. The n-Body Problem
1. The Two-Body Problem
1.1. Orbits
1.2. Anomalies
1.3. Collisions and Regularization
1.4. Geometry of the Kepler Problem
2. Collisions and Regularization
2.1. Necessary Conditions for Stability
2.2. Simultaneous Collisions
2.3. Binary Collisions
2.4. Singularities of Solutions in the n-Body Problem
3. Particular Solutions
3.1. Central Configurations
3.2. Homographic Solutions
3.3. The Amended Potential and Relative Equilibria
4. Final Motions in the Three-Body Problem
4.1. Classification of Final Motions According to Chazy
4.2. Symmetry of Past and Future
5. The Restricted Three-Body Problem
5.1. Equations of Motion. The Jacobi Integral
5.2. Relative Equilibria and the Hill Region
5.3. Hill''s Problem
6. Ergodic Theorems in Celestial Mechanics
6.1. Stability in the Sense of Poisson
6.2. Probability of Capture
Chapter 3. Symmetry Groups and Reduction Lowering the Order
1. Symmetries and Linear First Integrals
1.1. E. Noether''s Theorem
1.2. Symmetries in Nonholonomic Mechanics
1.3. Symmetries in Vakonomic Mechanics
1.4. Symmetries in Hamiltonian Mechanics
2. Reduction of Systems with Symmetry
2.1. Lowering the Order the Lagrangian Aspect
2.2. Lowering the Order the Hamiltonian Aspect
2.3. Examples: Free Motion of a Rigid Body and the Three-Body
Problem
3. Relative Equilibria and Bifurcations of Invariant Manifolds
3.1. Relative Equilibria and the Amended Potential
3.2. Invariant Manifolds, Regions of Possible Motions, and
Bifurcation Sets
3.3. The Bifurcation Set in the Planar Three-Body Problem
3.4. Bifurcation Sets and Invariant Manifolds in the Motion of a
Heavy Rigid Body with a Fixed Point
Chapter 4. Integrable Systems and Integration Methods
1. Brief Survey of Various Approaches to the Integrability of
Hamiltonian Systems
1.1. Quadratures
1.2. Complete Integrability
1.3. Normal Forms
2. Completely Integrable Systems
2.1. Action-Angle Variables
2.2. Noncommutative Sets of First Integrals
2.3. Examples of Completely Integrable Systems
3. Some Methods of Integrating Hamiltonian Systems
3.1. Method of Separation of Variables
3.2. Method of L-A Lax Pairs
4. Nonholonomic Integrable Systems
4.1. Differential Equations with Invariant Measure
4.2. Some Solved Problems of Nonholonomic Mechanics
Chapter 5. Perturbation Theory for Integrable Systems
1. Averaging of Perturbations
1.1. The Averaging Principle
1.2. Procedure for Eliminating Fast Variables in the Absence of
Resonances
1.3. Procedure for Eliminating Fast Variables in the Presence of
Resonances
1.4. Averaging in Single-Frequency Systems
1.5. Averaging in Systems with Constant Frequencies
1.6. Averaging in Nonresonant Domains
1.7. The Effect of a Single Resonance
1.8. Averaging in Two-Frequency Systems
1.9. Averaging in Multi-Frequency Systems
2. Averaging in Hamiltonian Systems
2.1. Application of the Averaging Principle
2.2. Procedures for Eliminating Fast Variables
3. The KAM Theory
3.1. Unperturbed Motion. Nondegeneracy Conditions
3.2. Invariant Tori of the Perturbed System
3.3. Systems with Two Degrees of Freedom
3.4. Diffusion of Slow Variables in Higher-Dimensional Systems,
and its Exponential Estimate
3.5. Variants of the Theorem on Invariant Tori
3.6. A Variational Principle for Invariant Toff. Cantori
3.7. Applications of the KAM Theory
4. Adiabatic Invariants
4.1. Adiabatic Invariance of the Action Variable in Single-
Frequency Systems
4.2. Adiabatic Invariants of Multi-Frequency Hamiltonian Systems
4.3. Procedure for Eliminating Fast Variables. Conservation Time
of Adiabatic Invariants
4.4. Accuracy of the Conservation of Adiabatic Invariants
4.5. Perpetual Conservation of Adiabatic Invariants
Chapter 6. Nonintegrable Systems
1. Near-Integrable Hamiltonian Systems
1.1. Poincare''s Methods
1.2. Creation of Isolated Periodic Solutions is an Obstruction to
Integrability
1.3. Applications of Poincare''s Method
2. Splitting of Asymptotic Surfaces
2.1. Conditions for Splitting
2.2. Splitting of Asymptotic Surfaces is an Obstruction to
Integrability
2.3. Applications
3. Quasi-Random Oscillations
3.1. The Poincare Map
3.2. Symbolic Dynamics
3.3. Nonexistence of Analytic First Integrals
4. Nonintegrability in the Neighborhood of an Equilibrium Position
Siegel''s Method
5. Branching of Solutions and Nonexistence of Single-Valued First
Integrals
5.1. Branching of Solutions is an Obstruction to Integrability
5.2. Monodromy Groups of Hamiltonian Systems with Single-
Valued First Integrals
6. Topological and Geometrical Obstructions to Complete
Integrability of Natural Systems with Two Degrees of Freedom
6.1. Topology of the Configuration Space of Integrable Systems
6.2. Geometrical Obstructions to Integrability
Chapter 7. Theory of Small Oscillations
1. Linearization
2. Normal Forms of Linear Oscillations
2.1. Normal Form of Linear Natural Lagrangian Systems
2.2. The Rayleigh-Fischer-Courant Theorems on the Behavior of
Characteristic Frequencies under an Increase in Rigidity and
under Imposition of Constraints
2.3. Normal Forms of Quadratic Hamiltonians
3. Normal Forms of Hamiltonian Systems Near Equilibria
3.1. Reduction to Normal Form
3.2. Phase Portraits of Systems with Two Degrees of Freedom in
the Neighborhood of an Equilibrium Position under Resonance
3.3. Stability of Equilibria in Systems with Two Degrees of Freedom
under Resonance
4. Normal Forms of Hamiltonian Systems Near Closed Trajectories
4.1. Reduction to the Equilibrium of a System with Periodic
Coefficients
4.2. Reduction of Systems with Periodic Coefficients to Normal
Form
4.3. Phase Portraits of Systems with two Degrees of Freedom Near
a Closed Trajectory under Resonance
5. Stability of Equilibria in Conservative Fields
Comments on the Bibliography
Recommended Reading
Bibliography
Index