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Mathematical Modeling of Diverse Phenomena NASA SP-437 dW = a••n c mathematical modeling Ok' of diverse phenomena james c. Howard NASA NASA SP-437 mathematical modeling of diverse phenomena James c. howard fW\SA National Aeronautics and Space Administration Scientific and Technical Information Branch 1979 Library of Congress Cataloging in Publication Data Howard, James Carson. Mathematical modeling of diverse phenomena. (NASA SP ; 437) Includes bibliographical references. 1, Calculus of tensors. 2. Mathematical models. I. Title. II. Series: United States. National Aeronautics and Space Administration. NASA SP ; 437. QA433.H68 515'.63 79-18506 For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 Stock Number 033-000-00777-9 PREFACE This book is intended for those students, 'engineers, scientists, and applied mathematicians who find it necessary to formulate models of diverse phenomena. To facilitate the formulation of such models, some aspects of the tensor calculus will be introduced. However, no knowledge of tensors is assumed. The chief aim of this calculus is the investigation of relations that remain valid in going from one coordinate system to another. The invariance of tensor quantities with respect to coordinate transformations can be used to advantage in formulating mathematical models. As a consequence of the geometrical simplification inherent in the tensor method, the formulation of problems in curvilinear coordinate systems can be reduced to series of routine operations involving only summation and differentia- tion. When conventional methods are used, the form which the equations of mathematical physics assumes depends on the coordinate system used to describe the problem being studied. This dependence, which is due to the practice of expressing vectors in terms of their physical components, can be removed by the simple expedient.of expressing all vectors in terms of their tensor components. For the benefit of those who have access to digital computers equipped with formula manipulation compilers, the convenience of computerized formulations will be demonstrated. No programming experience is necessary, and the few program- ming steps required will be explained as they occur. The first chapter is concerned with those aspects of the tensor calculus that are considered necessary for an understanding of later chapters. It is assumed that the reader has a knowledge of elementary vector analysis and matrix operations. In writing this part, I was influenced by the work of I. S. Sokolnikoff (ref. 1) and A. P. Wills (ref. 2). The definition of a tensor of rank r associated with a point of an N dimensional space, as an r linear form in the base vectors associated with the point, is due to Wills. I feel that this approach to tensor calculus will have a greater appeal to applied mathematicians than the conventional method of defining tensors in terms of their transformation laws. m JAMES C. HOWARD The reader may encounter unfamiliar entities such as covariant and contravariant vectors and tensors, and unfamiliar operations such as covariant differentiation. It will be seen, however, that the only operations involved in applying these concepts to practical problems are summation, in accordance with the summation convention, and differentiation. In using tensor methods to formulate mathematical models, considerable insight is obtained and the striking similarity of all formulations of physical systems becomes apparent. This is due to the fact that all such formulations evolve from a fundamental metric which is simply an expression for the square of the distance between two adjacent points on a surface. Hence, in addition to its utility, the method advocated has a definite educational value. As I. S. Sokolnikoff has noted, the best evidence of the remarkable effectiveness of the tensor apparatus in the study of nature is the fact that it is possible to include, between the covers of a small volume, a large amount of material that is of interest to mathematicians, physicists, and engineers. The major part of the book is devoted to applications using the theory given in the first chapter. The applications are chosen to demonstrate the feasibility of combining tensor methods and computer capability to formulate problems of interest to students, engineers, and theoretical physicists. Chapter 2 is devoted to aeronautical applications that culminate in the formulation of a mathematical model of an aeronautical system. In using chapter 2, only first- and second-order transfor- mations are required; the necessary theory is contained in the first 11 sections of chapter 1. In chapter 3, the equations of motion of a particle are formulated in tensor form. These formulations require an understanding of the Christoffel sym- bols (ref. 3) of the first and second kinds and the concept of covariant differentia- tion. The corresponding theory is contained in sections 1.11 and 1.12 of chapter 1. The methods described in chapter 4 can be used to formulate mathematical models involving fluid dynamics. An understanding of this chapter also requires a knowledge of the Christoffel symbols and covariant differentiation as described in sections 1.11 and 1.12 of chapter 1. The tensor theory contained in sections 1.13 through 1.18 of chapter 1, is required to formulate the cosmological models described in chapter 5. The final chapter describes how. the symbol manipulation language MACSYMA (ref. 4) may be used to assist in the formulation of mathematical models. The techniques described in this book represent an attempt to simplify the formulation of mathematical models by reducing the modeling process to a series of routine operations, which can be performed either manually or by computer. This attempt is part of a continuing effort in support of simulation experimentation in the Simulation Sciences Division. IV CONTENTS Page PREFACE iii 1. SCALARS, VECTORS, AND TENSORS 1 1.1 Summation Convention 3 1.2 Tensors 6 1.3 Physical Examples 11 1.4 Transformation Laws . 25 1.5 Base Vectors 29 1.6 Vector Transformations 31 1.7 Raising and Lowering of Indices 32 1.8 Bivector Transformations 34 1.9 Physical Components 36 1.10 Tensor Components - 39 1.11 Algebra of Tensors 40 1.12 Vector Derivatives and the Christoffel Symbols 47 1.13 Special Coordinate Systems .56 1.14 The Differential Operator V 60 1.15 The Riemann-Christoffel Tensor and the Ricci Tensor 64 1.16 The Ricci Tensor 69 1.17 Curvature Tensor 70 1.18 Einstein Tensor 72 1.19 References 74 Page 2. AERONAUTICAL APPLICATIONS 77 2.1 Transformations and Transformation Laws 79 2.2 Aeronautical Reference Systems 80 2.3 Transformation Law for Static Forces and Moments 82 2.4 Transformation Law for the Static-Stability Derivatives 85 2.5 Inverse Transformation Law for Static Forces and Moments. 86 2.6 Transformation by Computer 89 2.7 Transformation Law for Aerodynamic Stability Derivatives ...... 94 2.8 Computer Transformations of Aerodynamic Stability Derivatives. 103 2.9 Transformation of Moments and Products of Inertia 129 2.10 The Formulation of Mathematical Models of Aircraft 134 2.11 Aerodynamic Forces 136 2.12 Thrust Forces 142 2.13 Thrust Moments 144 2.14 Gravity Forces 146 2.15 Summation of Forces and Moments 153 .2.16 Spatial Orientation in Terms of Euler Angles 156 2.17 Spatial Orientation in Terms of Direction Cosines 159 2.18 Coordinates of the Aircraft in an Earth-Fixed Reference Frame ... 161 2.19 References 164 3. APPLICATIONS TO PARTICLE DYNAMICS 'l65 3.1 Formulation of Christoffel Symbols Using Coordinate Transformation Equations 167 3.2 Metric Tensor Inputs 172 3.3 The Velocity Vector 182 3.4 The Acceleration Vector 185 3.5 Equations of Motion in a General Curvilinear Coordinate System. 187 3.6 Computer Derivations of Equations of Motion of a Particle 193 3.7 Observations 206 3.8 Illustrative Examples 206 3.9 References - 216 vi Page 4. APPLICATIONS TO FLUID MECHANICS .~~. 217 4.1 Formulation of the Navier-Stokes Equations and the Continuity Equation 219 4.2 The Physical Form of the Navier-Stokes Equations 224 ' 4.3 Formulation of the Christoffel Symbols of the Second Kind 239 4.4 References 253 5. COSMOLOGICAL APPLICATIONS .~ 255 5.1 Newtonian and Relativistic Field Equations and Trajectory Equations 257 5.-21 Modeling Considerations 261 5.3 Anisotropic Model 263 5.4 Isotropic Model 269 5.5 Static Homogeneous Models 276 5.6 The Einstein Model Universe 277 5.7 The De Sitter Model 283 5.8 A Nonhomogeneous Case 284 5.9 Concluding Comments 287 5.10 References 287 6. ALTERNATIVE TECHNIQUES 289 6.1 REDUCE and MACSYMA. 291 6.2 Use of MACSYMA to Transform Aerodynamic Stability Derivatives 292 6.3 Printout of Input Commands 296 6.4 Central Processing Unit Times 297 6.5 Formulation of Christoffel's Symbols 300 6.6 Equations of Motion of a Particle 313 6.7 Formulating Models of Aeronautical Systems 317 6.8 Aeronautical Reference Systems 317 6.9 Transformation Equations 318 vn Page 6.10 Transformation Law for Static Forces 321 6.11 Transformation Law for Control Force Derivatives 322 6.12 Forces Produced by Linear Velocity Perturbations 324 6.13 Forces Produced by Angular Velocity Perturbations 327 6.14 Forces Produced by Linear Acceleration Perturbations 331 6.15 Gravity Forces 334 6.16 Inertia Forces 338 6.17 Resultant Forces 341 6.18 Special Forms of the Equations of Motion 349 6.19 Thrust Forces 358 6.20 Determination of the Geographical Location of Aircraft 360 6.21 Transformation Law for Static Moments 361 6.22 Transformation Law for Control Moment Derivatives 362 6.23 Moments Produced
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