## Differential Equations: Theory and Applications: theory and applications : with MappleThis 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 |

669 | |

675 | |

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### Common terms and phrases

ambient force angular assume asymptotically autonomous axis body canonical CD-ROM center of mass central force Chapter Christoffel symbol complex components conservation law constant constrained corresponding curvature cylinder defined definition denote derivative determine differential equations discussion domain dynamical systems easy eigenspaces eigenvalues eigenvectors equations of motion equivalent example exercise Existence and Uniqueness fixed points flow formula G Rn geometric given gives graph Hamiltonian system Hence hoop inequality inertia initial conditions integrable systems integral curve intersection inverse Jacobian matrix Jordan form level curves Liapunov function linear system linearly independent lst-order n x n neighborhood norm normal notation Note open set orbit orthogonal parameter particle phase portrait Picard iterates plane plot polar coordinates positive proof Proposition result satisfies sequence shown in Figure sketch solution stability submanifold subspace Suppose surface system x tangent transformation vector field velocity worksheet zero

### References to this book

Maple: Programming, Physical and Engineering Problems Victor Aladjev,Marijonas Bogdevicius Limited preview - 2006 |