Mathematical Methods of Classical Mechanics

Cover
Preface
Preface to the second edition
Translator's preface to the second edition
Contents
PART I NEWTONIAN MECHANICS
1 Experimental facts
1 The principles of relativity and determinacy
A Space and time
B Galileo's principle of relativity
C Newton's principle of determinacy
2 The galilean group and Newton's equations
A Notation
B Galilean structure
C Motion, velocity, acceleration
D Newton's equations
E Constraints imposed by the principle of relativity
3 Examples of mechanical systems
A Example 1: A stone falling to the earth
B Example 2: Falling from great height
C Example 3: Motion of a weight along a line under the action of a spring
D Example 4: Conservative systems
2 Investigation of the equations of motion
4 Systems with one degree of freedom
A Definitions
B Phase flow
C Examples
D Phaseflow
5 Systems with two degrees of freedom
A Definitions
B The law of conservation of energy
C Phase space
6 Conservative force fields
A Work of a force field along a path
B Conditions for a field to be conservative
C Central fields
7 Angular momentum
A The law of conservation of angular momentum
B Kepler's law
8 Investigation of motion in a central field
A Reduction to a one-dimensional problem
B Integration of the equation of motion
C Investigation of the orbit
D Central fields in which all bounded orbits are closed
E Kepler's problem
9 The motion of a point in three-space
A Conservative fields
B Central fields
C Axially symmetric fields
10 Motions of a system of n points
A Internal and external forces
B The law of conservation of momentum
C The law of conservation of angular momentum
D The law of conservation of energy
E Example: The two-body problem
11 The method of similarity
A Example
B A problem
PART II LAGRANGIAN MECHANICS
3 Variational principles
12 Calculus of variations
A Variations
B Extremals
C The Euler-Lagrange equation
D An important remark
13 Lagrange's equations
A Hamilton's principle of least action
B The simplest examples
14 Legendre transformations
A Definition
B Examples
C Involutivity
D Young's inequality
E The case of many variables
15 Hamilton's equations
A Equivalence of Lagrange's and Hamilton's equations
B Hamilton'sfunction and energy
C Cyclic coordinates
16 Liouville's theorem
A The phase flow
B Liouville's theorem
C Proof
D Poincare's recurrence theorem
E Applications of Poincare's theorem
4 Lagrangian mechanics on manifolds
17 Holonomic constraints
A Example
B Definition of a system with constraints
18 Differentiable manifolds
A Definition of a differentiable manifold
B Examples
C Tangent space
D The tangent bundle
E Riemannian manifolds
F The derivative map
19 Lagrangian dynamical systems
A Definition of a lagrangian system
B Natural systems
C Systems with holonomic constraints
D Procedure for solving problems with constraints
E Non-autonomous systems
20 E. Noether's theorem
A Formulation of the theorem
B Proof
C Examples
21 D'Alembert's principle
A Example
B Formulation of the D'Alembert-Lagrange principle
C The equivalence of the D' Alembert-Lagrange principle and the variational principle
D Remarks
5 Oscillations
22 Linearization
A Equilibrium positions
B Stability of equilibrium positions
C Linearization of a differential equation
D Linearization of a lagrangian system
E Small oscillations
23 Small oscillations
A A problem about pairs of forms
B Characteristic oscillations
C Decomposition into characteristic oscillations
D Examples
24 Behavior of characteristic frequencies
A Behavior of characteristic frequencies under a change in rigidity
B Behavior of characteristic frequencies under the imposition of a constraint
C Extremal properties of eigenvalues
25 Parametric resonance
A Dynamical systems whose parameters vary periodically with time
B The mapping at a period
C Linear mappings of the plane to itself which preserve area
D Strong stability
E Stability of an inverted pendulum with vertically oscillating point of suspension
6 Rigid bodies
26 Motion in a moving coordinate system
A Moving coordinate systems
B Motions, rotations, and translational motions
C Addition of velocities
D Angular velocity
E Transferred velocity
27 Inertial forces and the Coriolis force
A Coordinate systems moving by translation
B Rotating coordinate systems
28 Rigid bodies
A The configuration manifold of a rigid body
B Conservation laws
C The inertia operator
D Principal axes
29 Euler's equations. Poinsot's description of the motion
A Euler's equations
B Solutions of the Euler equations
C Poinsot's description of the motion
30 Lagrange's top
A Euler angles
B Calculation of the lagrangian function
C Investigation of the motion
31 Sleeping tops and fast tops
A Sleeping tops
B Fast tops
C A top in a weak field
D A rapidly thrown top
PART III HAMILTONIAN MECHANICS
7 Differential forms
32 Exterior forms
A 1-forms
B 2-forms
C k-forms
D The exterior product of two 1-forms
E Exterior monomials
33 Exterior multiplication
A Definition of exterior multiplication
B Properties of the exterior product
C Behavior under mappings
34 Differential forms
A Differential 1-forms
B The general form of a differential 1-form on R^n
C Differential k-forms
D The general form of a differential k-form on R^n
E Appendix. Differential forms in three-dimensional spaces
35 Integration of differential forms
A The integral of a 1-form along a path
B The integral of a k-form on oriented euclidean space R^k
C The behavior of differential forms under maps
D Integration of a k-form on an n-dimensional manifold
E Chains
F Example: the boundary of a polyhedron
G The integral of a form over a chain
36 Exterior differentiation
A Example: the divergence of a vector field
B Definition of the exterior derivative
C A theorem on exterior derivatives
D Stokes' formula
E Example 2 -- Vector analysis
F Appendix 1: Vector operations in triply orthogonal systems
G Appendix 2: Closed forms and cycles
H Appendix 3: Cohomology and homology
8 Symplectic manifolds
37 Symplectic structures on manifolds
A Definition
B The cotangent bundle and its symplectic structure
C Hamiltonian vector fields
38 Hamiltonian phase flows and their integral invariants
A Hamiltonian phase flows preserve the symplectic structure
B Integral invariants
C The law of conservation of energy
39 The Lie algebra of vector fields
A Lie algebras
B Vector fields and differential operators
C The Poisson bracket of vector fields
D The Jacobi identity
E A condition for the commutativity of flows
F Appendix: Lie algebras and Lie groups
40 The Lie algebra of hamiltonian functions
A The Poisson bracket of two functions
B The Jacobi identity
C The Lie algebras oj hamiltonian fields, hamiltonian junctions, and first integrals
D Locally hamiltonian vector fields
41 Symplectic geometry
A Symplectic vector spaces
B The symplectic basis
C The symplectic group
D Planes in symplectic space
E Symplectic structure and complex structure
42 Parametric resonance in systems with many degrees of freedom
A Symplectic matrices
B Symmetry of the spectrum of a symplectic transformation
C Stability
43 A symplectic atlas
A Symplectic coordinates
B Darboux's theorem
C Construction of the coordinates p1 and q1
D Construction of symplectic coordinates by induction on n
E Proof that the coordinates constructed are symplectic
9 Canonical formalism
44 The integral invariant of Poincare-Cartan
A A hydrodynamical lemma
B The multi-dimensional Stokes' lemma
C Hamilton's equations
D A theorem on the integral invariant of Poincare-Cartan
E Canonical transformations
45 Applications of the integral invariant of Poincare-Cartan
A Changes of variables in the canonical equations
B Reduction of order using the energy integral
C The principle of least action in phase space
D The principle of least action in the Maupertuis-Euler-Lagrange-Jacobi form
46 Huygens' principle
A Wave fronts
B The optical-mechanical analogy
C Action as a function of coordinates and time
D The Hamilton-Jacobi equation
47 The Hamilton-Jacobi method for integrating Hamilton's canonical equations
A Generating functions
B The Hamilton-Jacobi equation for generating functions
C Examples
48 Generating functions
A The generating function S_2(P, q)
B 2^n generating functions
C Infinitesimal canonical transformations
10 Introduction to perturbation theory
49 Integrable systems
A Liouville's theorem on integrable systems
B Beginning of the proof of Liouville's theorem
C Manifolds on which the action of the group R^n is transitive
D Discrete subgroups in R^n
50 Action-angle variables
A Description of action-angle variables
B Construction of action-angle variables in the case of one degree of freedom
C Construction of action-angle variables in R^2n
51 Averaging
A Conditionally periodic motion
B Space average and time average
C Proof of the theorem on averages
D Degeneracies
52 Averaging of perturbations
A Systems close to integrable ones
B The averaging principle
C Averaging in a single-frequency system
D Proof of the theorem on averaging
E Adiabatic invariants
F Proof of the adiabatic invariance of action
Appendix 1: Riemannian curvature
A Parallel translation on surfaces
B The curvature form
C The riemannian curvature of a surface
D Higher-dimensional parallel translation
E The curvature tensor
F Curvature in a two-dimensional direction
G Covariant differentiation
H The Jacobi equation
I Investigation of the Jacobi equation
J Geodesic flows on compact manifolds of negative curvature
K Other applications of exponential instability
Appendix 2: Geodesics of left-invariant metrics on Lie groups and the hydrodynamics of ideal fluids
A Notation: The adjoint and co-adjoint representations
B Left-invariant metrics
C Example
D Euler's equation
E Stationary rotations and their stability
F Riemannian curvature of a group with left-invariant metric
G Application to groups of diffeomorphisms
I Isovorticial fields
J Stability of planar stationary flows
K Riemannian curvature of a group of diffeomorphisms
L Discussion
Appendix 3: Symplectic structures on algebraic manifolds
A The hermitian structure of complex projective space
B The symplectic structure of complex projective space
C Symplectic structure on algebraic manifolds
Appendix 4: Contact structures
A Definition of contact structure
B Frobenius' integrability condition
C Nondegenerate fields of hyperplanes
D The manifold of contact elements
E Symplectification of a contact manifold
F Contact diffeomorphisms and vector fields
G Symplectification of contact diffeomorphisms and fields
H Darboux's theorem for contact structures
I Contact hamiltonians
J Computational formulas
K Legendre manifolds
L Contactification
M Integration of first-order partial differential equations
Appendix 5: Dynamical systems with symmetries
A Poisson action of Lie groups
B The reduced phase space
C Applications to the study of stationary rotations and bifurcations of invariant manifolds
Appendix 6: Normal forms of quadratic hamiltonians
A Notation
B Hamiltonians
C Nonremovable Jordan blocks
Appendix 7: Normal forms of hamiltonian systems near stationary points and closed trajectories
A Normalform of a conservative system near an equilibrium position
B Normal form of a canonical transformation near a stationary point
C Normal form of an equation with periodic coefficients near an equilibrium position
D Example: Resonance of order 3
E Splitting of separatrices
F Resonances of higher order
Appendix 8: Theory of perturbations of conditionally periodic motion, and Kolmogorov's theorem
A Unperturbed motion
B Invariant tori in a perturbed system
C Zones of instability
D Variants of the theorem on invariant tori
E Applications of the theorem on invariant tori and its generalizations
Appendix 9: Poincare's geometric theorem, its generalizations and applications
A Fixed points of mappings of the annulus to itself
B The connection betweenjixed points of a mapping and critical points of the generating function
C Symplectic diffeomorphisms of the torus
D Intersections of lagrangian manifolds
E Applications to determining fixed points and periodic solutions
F Invariance of generating functions
Appendix 10: Multiplicities of characteristic frequencies, and ellipsoids depending on parameters
A The manifold of ellipsoids of revolution
B Application to the study of oscillations of continuous media
C The effect of symmetries on the multiplicity of the spectrum
D The behavior of frequencies of a symmetric system under a variation of parameters preserving the symmetry
E Discussion
Appendix 11: Short wave asymptotics
A Quasi-classical approximation for solutions of Schrodinger's equation
B The Morse and Maslov indices
C Indices of closed curves
Appendix 12: Lagrangian singularities
A Singularities of smooth mappings of a surface onto a plane
B Singularities of projection of lagrangian manifolds
C Tables of normal forms of typical singularities of projections of lagrangian manifolds of dimension n <= 5
D Discussion of the normal forms
E Lagrangian equivalence
Appendix 13: The Korteweg-de Vries equation
Appendix 14: Poisson structures
A Poisson manifolds
B Poisson mappings
C Poisson structures in the plane
D Powers of volume forms
E The quasi-homogeneous case
F Varchenko's theorem
G Poisson structures and period mappings
Appendix 15: On elliptic coordinates
A Elliptic coordinates and confocal quadrics
B Magnetic analogues of the theorems of Newton and Ivory
Appendix 16: Singularities ofray systems
A Symplectic manifolds and ray systems
B Submanifolds of symplectic manifolds
C Lagrangian submanifolds in the theory of ray systems
D Contact geometry and systems of rays and wave fronts
E Applications of contact geometry to symplectic geometry
F Tangential singularities
G The obstacle problem
Bibliography of Symplectic Topology
Index