Mathematical Methods for Mechanics (eBook)

Preface 5
Contents 9
1 Mathematical Auxiliaries 17
1.1 Matrix Computations 17
1.2 Brief on Vector Analysis 33
1.3 Curves in R3 41
1.4 Linear Differential Equations 43
1.5 Linear Differential Systems of First Order 47
1.6 The Flux Integral and its Vector Field 54
1.7 Vector Spaces 61
1.8 Derivatives 68
1.9 Mappings in Banach Spaces 73
1.10 Convex Sets and Functions 77
1.11 Quadratic Functionals 86
2 Numerical Methods 94
2.1 Interpolation and Approximation 95
2.2 Orthogonal Polynomials 105
2.3 Numerical Integration 109
2.4 Initial Value Problems 120
2.5 Boundary Value Problems 143
2.6 Periodic Problems 148
2.7 Differential-Algebraic Problems 151
2.8 Hints to the MATLAB programs 156
3 Optimization 158
3.1 Minimization of a Function 159
3.2 Extrema with Constraints 164
3.3 Linear Programming 169
3.4 Linear-Quadratic Problems 179
3.5 Nonlinear Optimization 184
3.6 A Brief on Lagrange Theory 192
3.7 Hints to the MATLAB Programs 206
4 Variation and Control 208
4.1 Variation 209
4.2 Control Problems without Constraints 226
4.3 Control Problems with Constraints 235
4.4 Examples 241
4.5 On the Reentry Problem 251
4.6 Hints to the MATLAB programs 255
5 The Road as Goal 256
5.1 Bifurcation Problems 257
5.2 Scaling 272
5.3 Calculation of Singular Points 279
5.4 Ordinary Differential Systems 283
5.5 Hopf Bifurcation 290
5.6 Numerical Bifurcation 303
5.7 Continuation 310
5.8 Hints to the MATLAB Programs 315
6 Mass Points and Rigid Bodies 316
6.1 The Force and its Moment 316
6.2 Dynamics of a Mass Point 318
6.3 Mass Point in Central Field 325
6.4 Systems of Mass Points 334
6.5 The Three-Body Problem 343
6.6 Rotating Frames 349
6.7 Inertia Tensor and Top 354
6.8 On Multibody Problems 364
6.9 On Some Principles of Mechanics 368
6.10 Hints to the MATLAB Programs 372
7 Rods and Beams 373
7.1 Bending Beam 373
7.2 Eigenvalue Problems 381
7.3 Numerical Approximation 387
7.4 Frameworks of Rods 390
7.5 Frameworks of Beams 396
7.6 Hints to the MATLAB Programs 400
8 Continuum Theory 401
8.1 Deformations 401
8.2 The Three Transport Theorems 407
8.3 Conservation Laws 410
8.4 Material Forms 417
8.5 Linear Elasticity Theory 422
8.6 Discs 427
8.7 Kirchhoff’s Plate 429
8.8 Von Karman’s Plate and the Membrane 435
8.9 On Fluids and Gases 438
8.10 Navier-Stokes Equations 441
9 Finite Elements 449
9.1 Elliptic Boundary Value Problems 449
9.2 From Formula to Figure, Example 453
9.3 Constructing Finite Elements 459
9.4 Further Topics 466
9.5 On Singular Elements 481
9.6 Navier-Stokes Equations 485
9.7 Mixed Applications 496
9.8 Examples 503
9.9 Hints to MATLAB Programs 512
10 A Survey on Tensor Calculus 517
10.1 Tensor Algebra 517
10.2 Algebra of Alternating Tensors 534
10.3 Differential Forms in Rn 539
10.4 Tensor Analysis 551
10.5 Examples 564
10.6 Transformation Groups 569
11 Case Studies 574
11.1 An Example of Gas Dynamics 574
11.2 The Reissner-Mindlin Plate 576
11.3 Examples of Multibody Problems 578
11.4 Dancing Discs 581
11.5 Buckling of a Circular Plate 587
12 Appendix 590
12.1 Notations and Tables 590
12.2 Matrix Zoo 594
12.3 Translation and Rotation 596
12.4 Trigonometric Interpolation 598
12.5 Further Properties of Vector Spaces 604
12.6 Cycloids 606
12.7 Quaternions and Rotations 609
References 612
Index 623