Mathematical Aspects of Classical and Celestial Mechanics (Google eBook)

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Springer Science & Business Media, Jul 5, 2007 - Science - 531 pages
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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. From the reviews of the previous editions: '... The book accomplishes the goals it has set for itself. While it is not an introduction to the field, it is an excellent overview. ...' American Mathematical Monthly, Nov. 1989.
  

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Contents

523 Examples of Completely Integrable Systems
185
53 Some Methods of Integration of Hamiltonian Systems
191
532 Method of LA Pairs
197
54 Integrable NonHolonomic Systems
199
542 Some Solved Problems of NonHolonomic Mechanics
202
Perturbation Theory for Integrable Systems
207
612 Procedure for Eliminating Fast Variables NonResonant Case
211
613 Procedure for Eliminating Fast Variables Resonant Case
216

124 Poincares Equations
23
125 Motion with Constraints
26
13 Hamiltonian Mechanics
30
132 Generating Functions
33
133 Symplectic Structure of the Cotangent Bundle
34
134 The Problem of n Point Vortices
35
135 Action in the Phase Space
37
136 Integral Invariant
38
137 Applications to Dynamics of Ideal Fluid
40
14 Vakonomic Mechanics
41
141 Lagranges Problem
42
142 Vakonomic Mechanics
43
143 Principle of Determinacy
46
144 Hamiltons Equations in Redundant Coordinates
47
15 Hamiltonian Formalism with Constraints
48
152 Duality
50
16 Realization of Constraints
51
162 Holonomic Constraints
52
163 Anisotropic Friction
54
164 Adjoint Masses
55
165 Adjoint Masses and Anisotropic Friction
58
166 Small Masses
59
The nBody Problem
61
212 Anomalies
67
213 Collisions and Regularization
69
214 Geometry of Keplers Problem
71
22 Collisions and Regularization
72
222 Simultaneous Collisions
73
223 Binary Collisions
74
224 Singularities of Solutions of the nBody Problem
78
23 Particular Solutions
79
232 Homographic Solutions
80
233 Effective Potential and Relative Equilibria
82
24 Final Motions in the ThreeBody Problem
83
242 Symmetry of the Past and Future
84
25 Restricted ThreeBody Problem
86
252 Relative Equilibria and Hill Regions
87
253 Hills Problem
88
26 Ergodic Theorems of Celestial Mechanics
92
262 Probability of Capture
94
27 Dynamics in Spaces of Constant Curvature
95
272 Keplers Laws 177 343
96
273 Celestial Mechanics in Spaces of Constant Curvature
97
274 Potential Theory in Spaces of Constant Curvature
98
Symmetry Groups and Order Reduction
103
312 Symmetries in NonHolonomic Mechanics
107
313 Symmetries in Vakonomic Mechanics
109
314 Symmetries in Hamiltonian Mechanics
110
32 Reduction of Systems with Symmetries
111
322 Order Reduction Hamiltonian Aspect
116
Free Rotation of a Rigid Body and the ThreeBody Problem
122
33 Relative Equilibria and Bifurcation of Integral Manifolds
126
332 Integral Manifolds Regions of Possible Motion and Bifurcation Sets
128
333 The Bifurcation Set in the Planar ThreeBody Problem
130
334 Bifurcation Sets and Integral Manifolds in the Problem of Rotation of a Heavy Rigid Body with a Fixed Point
131
Variational Principles and Methods
134
41 Geometry of Regions of Possible Motion
136
412 Geometry of a Neighbourhood of the Boundary
139
413 Riemannian Geometry of Regions of Possible Motion with Boundary
140
42 Periodic Trajectories of Natural Mechanical Systems
145
422 Librations in NonSimplyConnected Regions of Possible Motion
147
423 Librations in Simply Connected Domains and Seiferts Conjecture
150
424 Periodic Oscillations of a MultiLink Pendulum 325
153
43 Periodic Trajectories of NonReversible Systems
156
432 Applications of the Generalized Poincare Geometric Theorem
159
44 Asymptotic Solutions Application to the Theory of Stability of Motion
161
441 Existence of Asymptotic Motions
162
442 Action Function in a Neighbourhood of an Unstable Equilibrium Position
165
443 Instability Theorem
166
444 MultiLink Pendulum with Oscillating Point of Suspension
167
445 Homoclinic Motions Close to Chains of Homoclinic Motions
168
Integrable Systems and Integration Methods
171
512 Complete Integrability
174
513 Normal Forms
176
52 Completely Integrable Systems
179
522 NonCommutative Sets of Integrals
183
614 Averaging in SingleFrequency Systems
217
615 Averaging in Systems with Constant Frequencies
226
616 Averaging in NonResonant Domains
229
618 Averaging in TwoFrequency Systems
237
619 Averaging in MultiFrequency Systems
242
6110 Averaging at Separatrix Crossing
244
62 Averaging in Hamiltonian Systems
256
622 Procedures for Eliminating Fast Variables
265
63 KAM Theory
273
632 Invariant Tori of the Perturbed System
274
633 Systems with Two Degrees of Freedom
279
634 Diffusion of Slow Variables in Multidimensional Systems and its Exponential Estimate
286
635 Diffusion without Exponentially Small Effects
292
636 Variants of the Theorem on Invariant Tori
294
637 KAM Theory for LowerDimensional Tori
297
638 Variational Principle for Invariant Tori Cantori
307
639 Applications of KAM Theory
311
64 Adiabatic Invariants
314
642 Adiabatic Invariants of MultiFrequency Hamiltonian Systems
323
643 Adiabatic Phases
326
644 Procedure for Eliminating Fast Variables Conservation Time of Adiabatic Invariants
332
645 Accuracy of Conservation of Adiabatic Invariants
334
646 Perpetual Conservation of Adiabatic Invariants
340
647 Adiabatic Invariants in Systems with Separatrix Crossings
342
NonIntegrable Systems
351
711 The Poincare Method
352
712 Birth of Isolated Periodic Solutions as an Obstruction to Integrability
354
713 Applications of Poincares Method
358
72 Splitting of Asymptotic Surfaces
360
722 Splitting of Asymptotic Surfaces as an Obstruction to Integrability
366
723 Some Applications
370
73 QuasiRandom Oscillations
373
731 Poincare Return Map
375
732 Symbolic Dynamics
378
733 Absence of Analytic Integrals
380
74 NonIntegrability in a Neighbourhood of an Equilibrium Position Siegels Method
381
75 Branching of Solutions and Absence of SingleValued Integrals
385
752 Monodromy Groups of Hamiltonian Systems with SingleValued Integrals
388
76 Topological and Geometrical Obstructions to Complete Integrability of Natural Systems
391
761 Topology of Configuration Spaces of Integrable Systems
392
762 Geometrical Obstructions to Integrability
394
763 Multidimensional Case
396
Theory of Small Oscillations
400
82 Normal Forms of Linear Oscillations
402
822 RayleighFisherCourant Theorems on the Behaviour of Characteristic Frequencies when Rigidity Increases or Constraints are Imposed
403
823 Normal Forms of Quadratic Hamiltonians
404
83 Normal Forms of Hamiltonian Systems near an Equilibrium Position
406
832 Phase Portraits of Systems with Two Degrees of Freedom in a Neighbourhood of an Equilibrium Position at a Resonance
409
833 Stability of Equilibria of Hamiltonian Systems with Two Degrees of Freedom at Resonances
416
84 Normal Forms of Hamiltonian Systems near Closed Trajectories
417
842 Reduction of a System with Periodic Coefficients to Normal Form
418
843 Phase Portraits of Systems with Two Degrees of Freedom near a Closed Trajectory at a Resonance
419
85 Stability of Equilibria in Conservative Fields
422
852 Influence of Dissipative Forces
426
853 Influence of Gyroscopic Forces
427
Tensor Invariants of Equations of Dynamics
431
912 Integral Invariants
433
913 PoincareCartan Integral Invariant
436
92 Invariant Volume Forms
438
922 Condition for the Existence of an Invariant Measure
439
923 Application of the Method of Small Parameter
442
93 Tensor Invariants and the Problem of Small Denominators
445
932 Application to Hamiltonian Systems
446
933 Application to Stationary Flows of a Viscous Fluid
449
94 Systems on ThreeDimensional Manifolds
451
95 Integral Invariants of the Second Order and Multivalued Integrals
455
96 Tensor Invariants of QuasiHomogeneous Systems
457
962 Conditions for the Existence of Tensor Invariants
459
97 General Vortex Theory
461
972 Multidimensional Hydrodynamics
463
973 Invariant Volume Forms for Lambs Equations
465
Recommended Reading
471
Bibliography
475
Index of Names
507
Subject Index
511
Copyright

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Page 476 - Conditions for the applicability, and estimate of the error, of an averaging method for systems which pass through states of resonance in the course of their evolution, Dokl.
Page 488 - Captures into resonance and scattering on resonance in dynamics of a charged relativistic particle in magnetic field and electrostatic wave Physica D 141.
Page 504 - In: Kozlov, VV (ed.) Dynamical Systems in Classical Mechanics. Transl., Ser. 2, Am. Math. Soc. 168(25) (1995), 91-128 587.
Page 493 - The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies," Planetary and Space Science, Volume 9, Issue 1 0, pp.
Page 478 - An approach to Arnol'd's diffusion through the calculus of variations. Nonlinear Anal., 26(6):1115-1135, 1996.
Page 479 - Homoclinic orbits to invariant tori of Hamiltonian systems. In: Kozlov, VV (ed.) Dynamical systems in classical mechanics. Transl., Ser. 2, Am. Math. Soc. 168(25)., 21-90 (1995).
Page 487 - Capture into Resonance: An Extension of the Use of Adiabatic Invariants,
Page 501 - Wiggins, S.: KAM theory near multiplicity one resonant surfaces in perturbations of a-priori stable Hamiltonian systems. J. Nonlinear Sci.
Page 488 - On the normal behaviour of partially elliptic lowerdimensional tori of Hamiltonian systems. Nonlinearity 10, No.

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Arnol'd, Steklov Mathematical Institute

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