Tensor Calculus and Analytical Dynamics : A Classical Introduction to Holonomic and Nonholonomic Tensor Calculus; And Its Principal Applications to the - John G. Papastavridis

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Tensor Calculus and Analytical Dynamics

A Classical Introduction to Holonomic and Nonholonomic Tensor Calculus; And Its Principal Applications to the

By: John G. Papastavridis

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Hardcover | 18 December 1998 | Edition Number 1

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Hardcover
434 Pages

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24.3 x 16.3 x 3.9

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Tensor Calculus and Analytical Dynamics provides a concise, comprehensive, and readable introduction to classical tensor calculus - in both holonomic and nonholonomic coordinates - as well as to its principal applications to the Lagrangean dynamics of discrete systems under positional or velocity constraints. The thrust of the book focuses on formal structure and basic geometrical/physical ideas underlying most general equations of motion of mechanical systems under linear velocity constraints.
Written for the theoretically minded engineer, Tensor Calculus and Analytical Dynamics contains uniquely accessbile treatments of such intricate topics as:

tensor calculus in nonholonomic variables

Pfaffian nonholonomic constraints

related integrability theory of Frobenius
The book enables readers to move quickly and confidently in any particular geometry-based area of theoretical or applied mechanics in either classical or modern form.

Industry Reviews

"This is a deep and complete workin view of the fact that very few books in tensor analysis have appeared recently, the present monograph is one of the most up to datehighly recommended" -L.Y. Bahar in Applied Mechanics Review, September 1999 "The book is very carefully written and published, it should find many readers and young scientists who want to follow the beautiful Analytical Dynamics in research and teaching." -Professor Peter MaiBer Applied Mechanics Review, September 1999 "The material covered in the book is certainly of a high interest in spite of the long tradition accumulated in the past and a considerable number of textbooks published... the book is intended to be a concise introductory tool to theoretical dynamics and aims to provide the reader with a strong background on the geometrical concepts supporting the formulation of the general equations of motion of constrained mechanical systems... Papastavridis achieves his purpose in handing down a coherent and understandable presentation inclusive of some solved examples and problems... The volume represents a valuable contribution to the dissemination of knowledge in the field of theoretical mechanics and may be recommended for graduate students and researchers in engineering, physics, and applied mathematics." -Meccanica, vol. 35, #5, 2001

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Algebra Applied Mathematics Engineering & Technology Mechanical Engineering & Materials Mechanical Engineering Technology in General Maths for Engineers Materials Science Mechanics of Solids

Dynamics & Vibration Testing of Materials Geometry Differential & Riemannian Geometry

Preface	p. ix
Acknowledgments	p. xiii
Summary of Conventions, Notations, and Basic Formulae	p. xxiii
Tensor Calculus
Introduction and Background
Some History	p. 3
Aims of Tensor Calculus	p. 3
Tensors and Geometry	p. 4
Tensors and Physics	p. 5
Tensors and Mechanics	p. 6
Exterior Forms (Cartan Calculus)	p. 7
Some Algebra	p. 8
Indices and Their Order	p. 8
Index Conventions	p. 9
Symmetry and Antisymmetry	p. 11
Special Symbols	p. 11
The Kronecker Delta	p. 11
The Levi-Civita Permutation Symbols	p. 12
The Generalized Kronecker Delta	p. 13
The Generalized Permutation Symbols	p. 15
Linear Equations, Cramer's Rule	p. 18
Functional Determinants (Jacobians)	p. 18
Derivatives of Determinants	p. 19
Some Geometry	p. 20
Primitive Concepts	p. 20
Set	p. 20
Group (G)	p. 20
Space	p. 21
Algebraic Definition	p. 21
Examples	p. 22
Coordinate System(s) (CS)	p. 22
Manifold	p. 22
Examples	p. 23
Coordinate Transformation(s) (CT)	p. 23
Systems of Coordinates vs. Frames of Reference	p. 24
Successive CT, Group Property	p. 25
Admissible CT	p. 25
Invariance	p. 26
Entity, or Object, Invariance	p. 26
Form Invariance	p. 26
Manifold Orientation	p. 27
Tensor Calculus (TC)	p. 27
Subspaces in a Manifold; Curves, Surfaces, etc.	p. 28
Special Cases	p. 29
Tensor Algebra
Introduction: Affine and Euclidean, or Metric, Vector Spaces	p. 31
Vectors	p. 31
0. Sum	p. 31
0'. Product with a Scalar	p. 31
0". Scalar, or Dot, Product	p. 32
Affine and Euclidean Point Spaces	p. 32
Linear Independence, Dimension and Basis	p. 33
Local Basis, or Frame	p. 33
Vector Subspace of an X[subscript n]	p. 34
Complementary Vector Subspaces	p. 34
Vector Algebra in a Euclidean Vector Space	p. 34
Dot Product	p. 34
Covariant and Contravariant Components of a Vector	p. 35
Dual of (or to) a Basis	p. 36
Change of Basis	p. 36
Introduction to Coordinate Transformations--Affine/Rectilinear Coordinates	p. 37
General Curvilinear (Nonlinear) Coordinate Transformations (CT)	p. 40
Curvilinear Coordinates and Their Differentials	p. 40
Integrability, Holonomic Coordinates	p. 41
Tensor Definitions	p. 42
Tensors of Rank One (Vectors)	p. 42
Tensors of Rank Two	p. 44
General Tensors	p. 45
Relative Tensors (RT)	p. 46
Some Motivation for RTs	p. 47
Geometrical Objects (GO)	p. 49
Rectangular Cartesian Coordinates (RCC)	p. 49
Polar vs. Axial Vectors	p. 50
Tensors vs. Matrices	p. 51
Numerical Tensors	p. 51
Properties of Tensor Transformation	p. 53
Algebraic Operations with Tensors	p. 54
Symmetry Properties	p. 55
Outer (or Exterior, or Direct) Multiplication of Tensors	p. 60
Contraction of Tensors	p. 60
Inner Multiplication or Transvection of Tensors	p. 60
Quotient Rule (QR)	p. 61
QR for Relative, and General Tensors	p. 62
Conjugate (or Associated) Tensors	p. 62
The Metric Tensor	p. 62
The Fundamental (Covariant) Metric Tensor	p. 62
Definitions	p. 63
Conjugate (Contravariant) Metric Tensor	p. 64
Left-/Right-Handedness (Orientation) of a CS	p. 65
Raising and Lowering of Indices	p. 65
Mixed Metric Tensor	p. 66
Tensorial Form of Ordinary Vector Algebra	p. 67
Bases in Curvilinear Coordinates	p. 67
Special Cases	p. 69
Transformation of Basis Vectors	p. 70
Length of Elementary Displacement Vector	p. 71
Elementary Area	p. 71
Elementary Volume	p. 73
Permutation Tensors	p. 73
Components of Vectors in Various Bases	p. 74
Applications	p. 77
Physical Components of Vectors and Tensors	p. 80
On Direct, or Dyadic (Polyadic etc.), or Invariant, Representations of Tensors	p. 81
General Tensors	p. 81
Dyadics	p. 82
Introduction to Riemannian Spaces	p. 84
Riemannian Space, R[subscript n]	p. 84
Tangent (Point and Vector) Spaces	p. 85
Flatness and Curvature	p. 88
Linearly, or Affinely, Connected Manifolds, Integrability	p. 89
Differences Between Affine (Nonmetric) and Metric Spaces	p. 90
Tensor Analysis
Introduction	p. 93
Differentiation of Tensor Components	p. 94
The Christoffel Symbols	p. 97
Definitions	p. 97
Properties	p. 97
Successive Coordinate Transformations	p. 100
Antisymmetric Part of the Christoffels	p. 100
The Covariant Derivative (CD)	p. 101
Definitions, Theorems	p. 101
The Absolute, or Intrinsic, Derivative (AD)	p. 102
Definitions	p. 102
Some Properties/Theorems of CDs and ADs	p. 103
Some Vector Analysis in Tensor Notation	p. 105
Parallelism, Straight Lines	p. 108
On Geodesics	p. 109
(First) Proof of the Christoffel Transformation Equations (Equations 3.3.2e ff)	p. 109
Geometrical Interpretation of Christoffels; Affine Manifolds; Torsion	p. 110
Euclidean Manifolds	p. 110
Relation of Equations 3.8.5a and b with the Earlier Christoffel Definitions (Equations 3.3.1a and b)	p. 112
Special Case: Moving Orthonormal Basis	p. 112
Geometrical Interpretation of CDs and ADs	p. 113
Geometrical Meaning of dV[superscript k], d*V[superscript k], and DV[superscript k]	p. 114
General Linearly, or Affinely, Connected and Metric-Equipped Manifolds	p. 115
Fundamental Theorem of Riemannian Geometry	p. 119
Asymmetric Affinities, Torsion	p. 120
Geometrical Significance of Torsion of a Manifold	p. 123
Curvature of a Manifold: Geometrical Aspects	p. 125
Additional Derivations of the Riemann-Christoffel Tensor (R-C) and Path Dependence	p. 127
Exactness (or Perfect Differential) Conditions	p. 127
Via the Generalized Stokes' Theorem	p. 129
Curvature of a Manifold: Algebraic Aspects	p. 129
Symmetries-Antisymmetries of R-C	p. 131
Number of Independent Components of R-C	p. 131
Contraction(s) of R-C	p. 133
Riemannian, or Sectional, Curvature	p. 133
Curvature vs. Flatness (Riemannian vs. Euclidean Spaces)	p. 134
Closing Remarks	p. 135
Nonholonomic (NH) Tensor Algebra	p. 136
Introduction	p. 136
Nonholonomic Coordinates and Bases	p. 136
On Notation	p. 138
NH Metric Tensor	p. 139
NH Vectors and Tensors	p. 140
Mixed H-NH Transformations	p. 141
NH Tensor Differentiation; Object of Anholonomicity	p. 142
NH Differentiation	p. 142
Definition	p. 142
The Nonholonomicity (or Anholonomicity) Object	p. 143
Remarks on AO	p. 144
Additional Uses of the AO	p. 145
Other Expressions for AO	p. 145
The AO in Three-Dimensional Torsionless Space	p. 146
NH Tensor Analysis: The Transitivity Equations	p. 149
Basic Results	p. 149
Transformation of the [gamma] Terms	p. 150
Geometrical Interpretation of the Transitivity Equations	p. 152
NH Tensor Analysis: NH Affinities and Christoffels	p. 153
NH Basis Gradients and Affinities	p. 153
Properties of the NH Affinities	p. 154
NH Affinities in a Riemannian Space (i.e., L[subscript n right arrow] R[subscript n]	p. 154
Transformation of the NH Affinities	p. 156
The First-Kind NH Christoffel-Like Symbols and Their Properties	p. 158
NH Tensor Analysis: NH Covariant Derivative	p. 160
Nonholonomic Riemann-Christoffel Tensor	p. 162
Analytical Dynamics
Introduction to Analytical Dynamics
Fundamental Concepts	p. 169
Configuration Space	p. 173
Kinematics	p. 174
Kinetics	p. 175
Introduction to Constraints--Purpose of Analytical Mechanics (AM)	p. 176
Whence the Need for AM	p. 178
Particle on a Curve and on a Surface
Introduction	p. 181
Particle in Ordinary Space: General Coordinates	p. 181
Particle in Ordinary Space: Natural, or Intrinsic, Variables	p. 185
Particle on a Curve	p. 194
Particle on a Surface	p. 197
Introduction to Surfaces, Velocity	p. 197
Tensor Analysis on a Surface	p. 198
Curve on a Surface	p. 202
Acceleration	p. 206
Forces, Equations of Motion	p. 208
General n-Dimensional (Riemannian) Surfaces	p. 211
Riemannian Space (R[subscript n]) Inside a Euclidean Space (E[subscript N]: n [ N [ [infinity]	p. 211
Differences Between Euclidean and Riemannian Arc-Lengths	p. 215
Problem of Embedding, or Immersing	p. 215
Problem of Equivalence, or Integrability	p. 216
Perturbation of Trajectories in Configuration Space, and Their Stability	p. 218
The Perturbation Equation	p. 219
The Energy Integral	p. 221
Alternative Forms	p. 222
Normal (Nonisochronous) Perturbations	p. 223
Lagrangean Mechanics: Kinematics
Introduction	p. 233
Holonomic Constraints	p. 233
Basic Definitions, System (or Generalized) Coordinates	p. 233
Scleronomic vs. Rheonomic Constraints	p. 236
Additional Holonomic Constraints	p. 236
Geometrical Interpretation of Holonomic Constraints	p. 237
Configuration Space	p. 237
Extended Configuration Space	p. 237
Velocity, Admissible and Virtual Displacements, and Acceleration in Particle and Holonomic System Variables	p. 238
Nonholonomic Coordinates, Velocities, etc.	p. 241
Basic Definitions, Quasi-Coordinates	p. 241
Properties of Pfaffian Transformations	p. 243
Particle and System Kinematics in Quasi-Variables	p. 245
The Transitivity Equations	p. 247
Nonintegrability Conditions	p. 249
Comprehensive Examples and Problems on Rigid-Body Kinematics	p. 252
Additional Pfaffian Constraints	p. 263
Holonomicity vs. Nonholonomicity	p. 264
Scleronomicity vs. Rheonomicity	p. 265
Catastaticity vs. Acatastaticity	p. 265
Nonholonomic Constraints (Equations 6.6.1)	p. 265
Theorem of Frobenius	p. 271
Geometrical Interpretation of Pfaffian Constraints	p. 278
Degrees of Freedom Revisited, Accessibility	p. 280
Geometrical Interpretation of the Frobenius Conditions (Equation 6.7.5e)	p. 282
First Interpretation	p. 282
Second Interpretation	p. 282
Lagrangean Mechanics: Kinetics
Introduction	p. 285
The Fundamental Kinematico-Inertial Quantities	p. 285
Kinetic Energy	p. 285
Holonomic Variables	p. 286
Nonholonomic Variables	p. 287
Metric in Configuration/Event Space	p. 288
Acceleration	p. 290
Inertia Force: Holonomic Components and Holonomic Euler-Lagrange Operator	p. 292
Inertia Force: Nonholonomic Components and Nonholonomic Euler-Lagrange Operator	p. 293
First Derivation	p. 293
Second Derivation	p. 295
The Forces	p. 301
The Physical Synthesis: Lagrange's Principle(s), Equations of Motion	p. 303
Lagrange's Principle (LP)	p. 303
Principle of Relaxation of Constraints (PRC)	p. 307
Lagrangean Forms of the Equations of Motion	p. 307
Holonomic Variables	p. 307
Nonholonomic Variables	p. 310
Appellian Forms of the Equations of Motion	p. 314
Summary	p. 314
The Central Equation	p. 325
The Power, or Energy Rate, Equations	p. 328
Holonomic Variables	p. 328
Nonholonomic Variables	p. 330
Comprehensive Examples and Problems on Lagrangean Dynamics	p. 335
Bibliography and References	p. 371
Index (Principal Authors and Subjects)	p. 381
Table of Contents provided by Syndetics. All Rights Reserved.

Tensor Calculus and Analytical Dynamics

A Classical Introduction to Holonomic and Nonholonomic Tensor Calculus; And Its Principal Applications to the

At a Glance

Hardcover

Industry Reviews

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How to return your order

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