Differential Equations: Theory and Applications: theory and applications : with Mapple

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Springer Science & Business Media, 2001 - Computers - 680 pages
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This book provides a comprehensive introduction to the theory of ordinary differential equations with a focus on mechanics and dynamical systems as important applications of the theory. The text is written to be used in the traditional way (emphasis on the theory with the computer component as optional) or in a more applied way (emphasis on the applications and the computer material). The accompanying CD contains Maple worksheets to use in working the exercises and extending the examples. The disk also contains special Maple code for performing various tasks. In addition to its use in a traditional one- or two- (there is enough material for two) semester graduate course in mathematics, the book is organized to be used for interdisciplinary courses in applied mathematics, physics, and engineering. Researchers and professionals may also find the supplementary material on the disk on discrete dynamical systems, theory of iterated maps, and code for performing specific tasks on the disks particularly useful.
 

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Contents

Introduction
1
12 Vector Fields and Dynamical Systems
14
13 Nonautonomous Systems
23
14 Fixed Points
26
15 Reduction to 1st0rder Autonomous
27
16 Summary
32
Techniques Concepts and Examples
33
21 Eulers Numerical Method
34
Newtonian Mechanics
361
91 The NBody Problem
362
911 Fixed Points
365
912 Initial Conditions
366
914 Stability of Conservative Systems
374
92 Eulers Method and the Nbody Problem
384
921 Discrete Conservation Laws
392
93 The Central Force Problem Revisited
401

212 The Analytical View
36
22 Gradient Vector Fields
39
23 Fixed Points and Stability
45
24 Limit Cycles
51
25 The TwoBody Problem
55
251 Jacobi Coordinates
57
252 The Central Force Problem
59
26 Summary
72
Existence and Uniqueness The How Map
75
31 Picard Iteration
78
32 Existence and Uniqueness Theorems
82
33 Maximum Interval of Existence
92
34 The Flow Generated by a TimeDependent Vector Field
95
35 The How for Autonomous Systems
104
36 Summary
112
OneDimensional Systems
115
41 Autonomous OneDimensional Systems
116
411 Construction of the Flow for 1D Autonomous Systems
123
42 Separable Differential Equations
128
43 Integrable Differential Equations
135
44 Homogeneous Differential Equations
147
45 Linear and Bernoulli Differential Equations
151
46 Summary
155
Linear Systems
157
51 Existence and Uniqueness for Linear Systems
162
52 The Fundamental Matrix and the Flow
165
53 Homogeneous Constant Coefficient Systems
174
54 The Geometry of the Integral Curves
180
541 Real Eigenvalues
182
542 Complex Eigenvalues
192
55 Canonical Systems
211
551 Diagonalizable Matrices
214
552 Complex Diagonalizable Matrices
217
Jordan Forms
219
56 Summary
227
Linearization and Transformation
231
62 Transforming Systems of DEs
247
621 The Spherical Coordinate Transformation
253
622 Some Results on Differentiable Equivalence
257
63 The Linearization and Flow Box Theorems
266
Stability Theory
275
71 Stability of Fixed Points
276
72 Linear Stability of Fixed Points
279
721 Computation of the Matrix Exponential for Jordan Forms
280
73 Nonlinear Stability
290
74 Liapunov Functions
292
75 Stability of Periodic Solutions
303
Integrable Systems
323
81 First Integrals Constants of the Motion
324
82 Integrable Systems in the Plane
329
83 Integrable Systems In 3D
334
84 Integrable Systems in Higher Dimensions
348
931 Effective Potentials
404
932 Qualitative Analysis
405
933 Linearization and Stability
409
934 Circular Orbits
410
935 Analytical Solution
412
94 RigidBody Motions
424
941 The RigidBody Differential Equations
432
942 Kinetic Energy and Moments of Inertia
438
943 The Degenerate Case
446
944 Eulers Equation
447
945 The General Solution of Eiders Equation
451
Motion on a Submanifold
463
101 Motion on a Stationary Submanifold
464
1011 Motion Constrained to a Curve
471
1012 Motion Constrained to a Surface
476
102 Geometry of Submanifolds
484
103 Conservation of Energy
493
104 Fixed Points and Stability
495
105 Motion on a Given Curve
502
106 Motion on a Given Surface
513
1061 Surfaces of Revolution
520
1062 Visualization of Motion on a Given Surface
526
107 Motion Constrained to a Moving Submanifold
531
Hamiltonian Systems
541
111 1Dimensional Hamiltonian Systems
544
1111 Conservation of Energy
547
112 Conservation Laws and Poisson Brackets
551
113 Lie Brackets and Arnolds Theorem
565
1131 Arnolds Theorem
567
114 Liouvilles Theorem
582
Elementary Analysis
589
A2 The Chain Rule
595
A3 The Inverse and Implicit Function Theorems
596
A4 Taylors Theorem and The Hessian
602
A5 The Change of Variables Formula
606
Lipschitz Maps and Linearization
607
B1 Norms
608
B2 Lipsdiitz Functions
609
B3 The Contraction Mapping Principle
613
B4 The Linearization Theorem
619
Linear Algebra
633
C2 Bilinear Forms
636
C3 Inner Product Spaces
638
C4 The Principal Axes Theorem
642
C5 Generalized Eigenspaces
645
C6 Matrix Analysis
656
C61 Power Series with Matrix Coefficients
662
CDROM Contents
665
Bibliography
669
Index
675
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About the author (2001)

David Betounes if Professor of Mathematics at the University of Southern Mississippi.

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