THE HYPERCIRCLE IN MATHEMATICAL PHYSICS -...

THE HYPERCIRCLE IN

MATHEMATICAL PHYSICS

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Cambridge University Press978-1-107-66655-9 - The Hypercircle in Mathematical Physics: A Method for the Approximate Solution of Boundary Value ProblemsJ. L. SyngeFrontmatterMore information

http://www.cambridge.org/9781107666559

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THE HYPERCIRCLE IN

MATHEMATICAL PHYSICS

A METHOD FOR

THE APPROXIMATE SOLUTION OF

BOUNDARY VALUE PROBLEMS

BY

J. L. SYNGE, So.D., F.R.S. Senior Professor, School of Theoretical Physics

Dublin Institute for Advanced Studies

OAMBRIDGE

AT THE UNIVERSITY PRESS

1957






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© Cambridge University Press 1957

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First published 1957 First paperback edition 2011

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CONTENTS

Preface page xi

Introduction page 1

PART I. NO METRIC

Ohapter 1. GEOMETRY OF FUNCTION-SPACE WITHOUT A

METRIC

1·1. Introductory ideas page 7 Representation of numbers, p. 7. Representation of functions, p. 8. The idea of function-space, p. 9. Linear dependence and independence, p. II. The number of dimensions in function-space is infinite, p. II. Pictorial representation of function-space, p. 12. Physical examples: (i) Vibrating string, p. 13; (ii) A gas, p. 15. (iii) A condenser, p. 15; (iv) An elastic body, p. 17. Exercises, p. 17.

1·2. F-vectors page 17 P-space and F-space distinguished, p. 17. F-vectors defined, p. 18. Addition, subtraction and multiplication by a scalar, p. 19. Logic and intuition, p. 20. F-points and bound F-vectors, p. 20. The parallelogram law of addition, p. 22. Translational transformations, p. 23. Exercises p.23.

1·3. Straight lines and linear subspaces page 24

Straight lines, p. 24. Linear dependence of two F-vectors, p. 24. Linear dependence of n F-vectors, p. 25. Pictorial representation of functionspace, p. 26. Linear n-spaces, p. 27. Intersections of linear m-spaces and linear n-spaces, p. 29. Linear subspaces in general, p. 31. Is an F-vector nothing but a certain set of functions? p. 33. Exercises, p. 34.

PART II. POSITIVE-DEFINITE METRIC

Ohapter 2. GEOMETRY OF FUNCTION-SPACE WITH POSITIVE

DEFINITE METRIC

2·1. The scalar product and metric in F -space page 37

Conditions to be satisfied by the scalar product, p. 37. Examples of scalar products: (i) Hilbert scalar product. p. 38; (ii) Dirichlet scalar product, p. 38; (iii) Minkowski scalar product, p. 38. Differentiation of a scalar product, p. 39. Metric in F-space; definite, semi-definite and indefinite, p. 40. Scalar product derived from metric, p. 41. Admissible functions. p. 42. Exercises, p. 43.






vi CONTENTS

2·2. Length and angle in F-space page 44 Length and distance in F-space, p. 44. Example, p. 44. Unit F-vectors and normalization, p. 45. The Schwarz inequality, p. 45. Angles in F.space, p. 46. Elementary geometry of the triangle, p. 47. The triangle inequality, p. 49. Orthogonality, p. 49. Exercises, p. 50.

2·3. Orthonormal F-vectors page 50 Orthonormalityand linear independence, p. 50. Orthogonal transformations, p. 51. Orthogonal transformations in a linear 2-space, p. 52. The process of orthonormalization, p. 52. Examples of orthonormalization, p. 55. The extended Schwarz inequality, p. 59. The process of orthonormalization cannot fail, p. 60. Resolution of an F-vector in, and orthogonal to, a linear n-space L n , p. 60. Example, p. 62. The normal drawn to a linear n-space Ln and its minimum property, p. 62. Example: approximation by a step-function, p. 64. Orthogonal linear subspaces, p. 66. Exercises, p. 67.

2·4. Hyperplanes page 68

Hyperplanes of class 1 (HI)' p. 68. Hyperplanes of class n (Hn), p. 69. The normal drawn to a hyperplane Hn and its minimum property, p. 71. Orthogonality of a linear n-space Ln and a hyperplane Hn> p. 72. The non-orthogonality of hyperplanes, p. 74. Examples, p. 74. Exercises, p. 75.

2·5. Hyperspheres page 76 Neighbourhoods, p. 76. No closed boxes in function-space, p. 76. n-spheres and hyperspheres defined, p. 77. Bounds on X2 as X ranges on a hypersphere, p. 79. Bounds on X.G as X ranges on a hypersphere, p. 80. Examples, p. 81. Intersection of a hypersphere and a linear n-space, p. 83. Intersection of a hypersphere and a hyperplane, p. 84. Hyperplanes of class 1 tangent to a hypersphere, p. 86. Tangent hypercones, p. 87. Exercises, p. 88.

2·6. Hypercircles page 89 Hypercircles of class 1, p. 89. Hypercircles of class n, p. 90. Bounds on X2 as X ranges on a hypercircle, p. 93. Bounds on X.G as X ranges on a hypercircle, p. 94. Example, p. 95. Exercises, p. 96.

2·7. The key to the hypercircle method page 97 Linear subspaces and orthogonality, p. 98. The vertices V', V" of two nonintersecting orthogonal linear subspaces L', L", p. 99. Uniqueness theorem for vertices, p. 101. Inequalities and minimum principles for two non-intersecting orthogonal linear subspaces L', L", p. 101. The vertices V', V" of two non-intersecting orthogonal linear subspaces of finite dimensionality L;, L;, p. 102. The intersection of two orthogonal linear subspaces L', L", p. 105. Inequalities and minimum principles for two intersecting orthogonal linear subspaces L', L", p. 105. The method of the hypercircle, p. 107. The existence of the hypercircle and its equations, p. 107. Bounds involving S, p. 110. The case where L" contains the origin 0, p. 113.






CONTENTS Vll

Summary of formulae for application of the method of the hypercircle, p. U8. Use of non· linear subspaces, p. 121. Bounds on 8 2 obtained without solving exactly the equations for the vertices V', V", p. 121. Exercises, p. 124.

Ohapter 3. THE DIRICHLET PROBLEM FOR A FINITE DOMAIN

IN THE EUCLIDEAN PLANE

3·1. The Dirichlet problem in physics page 125 Statement of the Dirichlet problem, the Neumann problem, and the mixed problem, p. 125. Electrostatics, p. 128. Current flow in a conductor, p. 129. Heat conduction, p. 129. Irrotational fluid flow, p. 129. Flow of a viscous fluid through a tube, p. 130. Elastic membrane, p. 130. Torsion, p. 130. The conjugate harmonic function and multiple connectivity, p. 131. Exercises, p. 133.

3·2. Splitting the Dirichlet problem page 134 Function-space and scalar product, p. 134. The linear subspaces L', L", p. 135. Orthogonality of L' and L"; permissible discontinuities, p. 136. Case where the boundary value function f is only piecewise continuous, p. 137. To make L' or L n pass through 0, p. 137. Summary of splitting, p. 138. Examples of F-points in L' and L n , p. 139. Exercises, p. 140.

3·3. The hypercircle page 141 The hypercircle of class 1 and associated inequalities, p. 141. Example: (u)B=tr2 for a square, p. 143. The vertices V', vn and the hypercircle of class r+s, p. 145. Bounds on 8 2 and 8.G, p. 147. Examples (1) (U)B=Xn

for a square, p. 148; (2) Square condenser, p. 151. Exercises, p. 154.

3·4. Bounds for the solution and its derivatives at an interior point page 155

The GreIm's F.vector, p. 155. Bounds for the solution, p. 157. Checks, p. 159. Bounds for the first derivatives, p. 159. Checks, p. 160. Bounds for the second derivatives, p. 161. The nature of the calculations, p. 163. Example: deformed tubular condenser, p. 164. Exercises, p. 168.

3·5. Pyramid F-vectors page 168 Pyramid functions, p. 168. Pyramid F-vectors of the first class, p. 170. Pyramid F·vectors of the second class, p. 171. Triangulation and polyhedral graphs, p. 173. Approximation to a given function and its first derivatives by a linear combination of pyramid functions, p. 175. Plan for the use of pyramid F-vectors in the hypercircle method, p. 176. Summary of plan for the use of pyramid F·vectors, p. 178. The linear independence of T~ and linear dependence of T:, p. 179. The conjugate harmonic function, p. 180. Case of multiple connectivity, p. 182. Strip F·vectors, p. 184. Summary of procedure for the Dirichlet problem in a multiply connected plane region, using pyramid and strip F-vectors, p. 184. Linear independence of strip F-vectors, p. 185. Bounds on the mean value of the gradient of the solution in a triangle of the network, p. 185. Exercises, p. 188.






viii CONTENTS

3·6. Hexagonal pyramid F-vectors page 188 Definition and normalization, p. 188. Scalar products, p. 190. Determination of the vertices V', V R p. 191. The bisection process, p. 194. Example: ,

hexagonal condenser, p. 195. Exercises, p. 199.

3·7. Square pyramid F-vectors page 200 Definition and normalization, p. 200. Scalar products, p. 202. Determination of the vertices V', V R , p. 202. The bisection process, p. 203. Example: square condenser, p. 204. Exercises, p. 209.

3·8. Approximation by linear interpolation page 209 Exercise, p. 213.

Ohapter 4. THE TORSION PROBLEM

4·1. The torsion problem as a Neumann problem and as an extended Dirichlet problem page 214

The torsion problem stated, p. 214. Torsional rigidity, p. 216. Transformation of axes, p. 217. Exercises, p. 219.

4·2. The hypercircle method applied to the torsion problem page 219

The moment of inertia F-vector and the spin F-vector, p. 219. Splitting the Neumann problem, p. 220. Some inequalities for torsional rigidity, p. 220. Splitting the extended Dirichlet problem, p. 222. Summary of procedure for the torsion problem, applicable to a multiply connected section, p. 224. Approximation to stress and warping, p. 225. Torsion of a rectangle, p. 226. Example: torsion of a hollow square, p. 230. Exercises, p.233. .

4·3. Pyramid F-vectors in the torsion problem page 234 Ring F-vectors, p. 235. Pyramid F-vectors of the second class in the case of multiple connectivity, p. 236. Scalar products with S~, p. 237. Cases of broken pyramids, p. 238. Exercises, p. 238.

4·4. The torsion of a regular hexagon page 241 General plan, p. 241. Use of approximate solutions for the weights of the hexagonal pyramid F-vectors, p. 246. The multiplication process for quick convergence, p. 249. The approximation n= 1, p. 252. The approximation n=2, p. 253. The approximation n=4, p. 254. The approximation n=8, p. 258. The approximation n= 16, p. 261. Approximation for stress and warping, p. 262. Summary of results for the torsion of a regular hexagon, p. 267. Exercises, p. 269.

4·5_ The torsion of a hollow square page 270 General plan, p. 270. Use of approximate solutions for the weights of the F-vectors, p. 278. The multiplication process for quick convergence,p. 280. The approximation n=8, p. 283. The approximation n= 16, p. 286. Approximation for stress and warping, p. 289. Summary of results for the torsion of a hollow square, p. 291. Exercises, p. 291.






CONTENTS ix

Chapter 5. VARIOUS BOUNDARY VALUE PROBLEMS

5·1. Boundary value problems connected with variational principles page 292

A variational principle and corresponding differential equation for a single unknown, p. 292. Splitting the differential equation, p. 293. The scalar product, p. 293. The orthogonal linear subspaces, p. 294. The case of several unknowns, p. 295. More general case, p. 296. Exercises, p. 299.

5·2. Dirichlet-Neumann problems page 300 Pyramid F-vectors for three-dimensional P-space, p. 301. First mixed problem, p. 304. Transfer of boundary conditions from curvilinear to straight boundaries, p. 305. Second mixed problem, p. 307. Exercises, p. 311.

5·3. Examples of mixed boundary value problems page 312 First mixed problem: viscous flow in a channel, p. 312. Flow problem: the Dirichlet integral and flux through the channel, p. 314. Flow problem: flow through a channel of semicircular cross-section, p. 316. Flow problem: the approximation n=4; lower bound for 8 2, p. 319. Flow problem: the approximation n=4: upper bound for 8 2, p. 321. Flow problem: the approximation n= 8, p. 324. Second mixed problem: membrane with elastic support, p. 329. Membrane problem: first approximation, p. 332. Membrane problem: higher approximation, p. 333. Exercises, p. 335.

5·4. Equilibrium of an elastic body page 336 Equations of equilibrium and compatibility, p. 336. Splitting the problem, p. 337. Example: cube deformed by its own weight, p. 339. Example: beam bent by terminal forces, p. 343. Incompressible elastic bodies, p. 345. The Green's tensor of elasticity, p. 347. Bounds at a point in elastic equilibrium, p. 350. Minimum principles in elastic equilibrium, p. 354. Exercises, p. 354.

5·5. The biharmonic equation page 355 Biharmonic functions and their conjugates, p. 355. The biharmonic equation in hydrodynamics, p. 357. The biharmonic equation in elasticity, p.359. Splitting the biharmonic problem, p. 361. Example: viscous flow across a tank, p. 363. Example: plate under pressure, p. 365. Exercises, p.366.

P ART III. INDEFINITE METRIC

Chapter 6. GEOMETRY OF FUNCTION-SPACE WITH IN

DEFINITE METRIC

6·1. Null vectors, null cones, orthogonality, Minkowskian F -spaces page 371

Resume of available facts,p. 371. Null F-vectors, p. 372. Null cones, p. 372. Orthogonality and orthogonal projection, p. 373. Orthonormalization, p. 376. Minkowskian F.spaces, p. 379. Exercises, p. 380.






x CONTENTS

6·2. Hyperplanes, pseudohyperspheres, pseudohypercircles page 380

Hyperplanes, p. 380. Pseudohyperspheres, p. 381. Pseudohypercirc1es, p. 383. Bounds connected with PHS, p. 384. Exercises, p. 385.

6·3. The method of the pseudohypercircle page 385 The vertices V', V" of two non· intersecting linear subspaces L', L", p. 385. The vertices V', V" of two non-intersecting linear subspaces of finite dimensionality L;, L;, p. 386. The pseudohypercircle, p. 386. Stationary principles, p. 388. Exercises, p. 388.

6·4. Approximation by orthogonal projection in Minkowskian F-space page 389

The separation of two straight lines, p. 389. The separation of two linear subspaces of finite dimensionality L;, L;, p. 391. Plan for approximation, p. 391. Exercises, p. 392.

Chapter 7. VIBRATION PROBLEMS

7 ,1. Scalar vibrations page 394 The scalar wave equation, p. 394. Splitting the problem, p. 395. Eigenvalue problems, p. 396. Triangulation in membrane problems, p. 398. Exercises, p.402.

7·2. Elastic and electromagnetic vibrations page 402 The elastic wave equations, p. 402. Splitting elastic vibrations, p. 403. Stationary principles for elastic vibrations, p. 405. Maxwell's equations, p. 406. Splitting electromagnetic vibrations. p. 406. Stationary principles for electromagnetic vibrations, p. 408. Exercises, p. 409.

Note A. The torsion of a hollow square page 410

Note B. The Green's tensor or fundamental solution for the equilibrium of an anisotropic elastic body page 411

Bibliography page 414

Index page 419






PREFACE

This book describes a technique for the approximate solution of certain boundary value problems of mathematical physics. This technique involves concepts of function-space. These are developed ab initio, so that no special knowledge beyond the calculus is required on the part of the reader. Thus it is an elementary book in a mathematical sense, but the arguments are, I hope, mathematically exact, although I have tried to avoid the rather bleak axiomatics which repel mathematical physicists and engineers for whom the book is intended and for whom function-space will remain a means to an end and not an end in itself.

The book has been ten years in the making, during which time I have given isolated lectures, and in some cases short courses, on various aspects of the subject in a number of places: Princeton University, Massachusetts Institute of Technology, University of Leeds, Harvard University, Brown University, Carnegie Institute of Technology, Institute for Fluid Dynamics and Applied Mathematics (University of Maryland), Severi Jubilee Celebration (Rome, 1950), St Andrews Mathematical Colloquium (1951), University of Trieste, Henri Poincare Colloquium (Paris, 1954), and the Dublin Institute for Advanced Studies. The experience so gained has been valuable because it brought home to me how reluctant mathematicians and physicists are in this age of analysis to use geometrical intuition as a guide, particularly when the geometry is that of a multidimensional space, or (worse) a space with an infinity of dimensions. Since this intuitional approach seems to me the first essential (for suggestion, not for proof), I have developed it very slowly in the early part of the book in order to establish a common understanding with the reader.

Outside this special field of application my knowledge of function -space is slight, and I am much indebted to Dr F. Smithies, mathematical adviser of the Cambridge University Press, whose kindly advice has kept me from wandering too far from current usages.

As stated in the Introduction, Professor W. Prager was joint originator of the hypercircle method in 1946; my thanks are due to him not only for the stimulating collaboration at that time but also for reading most of the manuscript of this book. Professor A. J. McConnell, Provost of Trinity College, Dublin, has taken an active interest in the work, his approach through variational






xii PREFACE

principles (see Chapter 5) throwing light on the range of applicability of the method. Critical discussions with Professors J. B. Diaz and A. Weinstein have helped me very much, and Dr J. McMahon and Mr V. G. Hart, while Scholars at the Dublin Institute for Advanced Studies, gave me assistance in the preparation of the manuscript; the computations of Chapter 5 were done by Mr Hart, who also wrote more than a third of the text of that chapter.

Mr Hart has given invaluable assistance in proof-reading. In a book of this sort, a complete absence of errors is not to be expected; the most we can hope is that they are few and venial. Our task has been greatly lightened by the efficiency ofthe Cambridge University Press in handling a rather difficult job.

DUBLIN J.L.S. August 1956






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