General References for Volume 1 ,

[lJ V.I. Arnold, Mathematical methods of classical me­chanics, Springer Verlag, Berlin, and New York, 1978.

[2J R. Becker, Introduction to Theoretical Mechanics, McGraw-Hill ,New York~ 1954.

[3J Y. Choquet Bruhat, Problems and Solutions in Mathema­tical Physics, Holden Day, San Francisco, 1967.

[4J J.P. den Hartog, Mechanical Vibrations, McGraw-Hill, 1950.

[5J H. Goldstein, Classical Mechanics, Addison Wesley, Reading, Mass., 1950.

[6 J r~i kh 1 in, Linear Equations of Mathemati ca 1 Physics, Holt, Rinehart, and Winston, New York, 1967.

[7J P.M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953, Vol. I and II.

[8J N.I. Muskhelishvili, Some Basic Problems of the Ma­thematical Theory of Elasticitl, Noordhorf, Gronigen, Ho 11 and, 1953.

[9J W. Prager, Introduction to Mechanics of Continua, Ginn, & Co., Boston, 1961.

[10 J A.N. Tikhonov and A.A. Samarsk-ji. Equations of Mathe­matical Physics (in Russian), "Nauka", Moscow, 1966.

[llJ S. Timoshenko and J.N. Goodier, Theory of Elasticity, McGraw-Hill, New York, 1951.

366

GENERAL REFERENCES FOR VOLUME 1 367

[12 ]

[13 ]

[14 ]

[15 ]

[16 ]

[17J

[18 ]

[19 ]

K. Yosida, Lectures on differential and integral eguations, Intersc;ence, New York, 1960.

R. Courant and D. Hilbert, Methods of mathematical physics, Interscience, New York, vol. 1,1953, vol. 2 1962.

L. D. Landau and E.M. Lifshitz, Mechanics, Vol. 1 of Course i~.:t;heox.~Jtc~JQh.Ys i c~, Pergammon Press, Ox­ford, London and New York, 1 60.

S. G. Mikhlin, Mathematical physics - an advanced course, North Holland Publ. Co. Amsterdam, 1970.

F. Riesz and Sz. Nagy, Lectures on func­tional analysis, Akad. Krado, Budapest 1952, Ungar Publ. 1962, English translation.

S.L. Sobolev, Some applications of functional analysis to mathematical physics, Leningrad, Univ. 1945, English translation, A.M.S. monograph translation #7, Providence, Rh. I., 1963.

N. Dunford and J. Schwartz, Linear Operators, Vol. 1, Interscience, 1958.

Yosida, Functional Analysis, 2nd edition, Springer, New York, 1968.

Index

Abovskii, N.P. and Andreev, N.L., 240, 245 Abraham, R., 104 Absolute minimum of a function, 58 Action integral, 49, 117 Adams, R.A., 244, 247, 307 Admissible displacements and forces, 217 Akhiezer, N.J., 110, 191 Akhiezer, N.J. and Glazmann, I.M., 191, 364, 365 Alexiewicz, A. and Orlicz, W., 191, 348 Almansi, 257, 258, 259 Anderson, N. and Arthurs, A.M., 201 Anton, H. and Bahar, L.Y., 106 Argyris, J.H. and Kelsey, S., 204 Arnold, V.I., 366 Aronszajn, N., 298, 309 Arthurs, A.M., 146, 190, 191, 198, 204, 242, 331,

338,·339, 340 Arthurs, A.M., monograph on dual variational principles, Ill, 146, 191, 331, 333

Ashby, N.W., Brittain, E., Love, W.F. and Wyss, W., 70, 105

Attiya-Singer index theory, 1 Aubin, J.P., 243, 246, 307 Averages of energy forms and natural frequencies,

3, 6 Averaging techniques in plate theory, 229

Bahar, L.Y., 106 Banach, 114, 115, 118, 132, 133, 154, 161, 162,

167, 279, 341, 349(all refer to Banach space arguments)

Banichuk, N.V., 226 Barnes, E.R., 198 Bateman's variational principle in fluid flow,179 Baumeister, R., 204 Bazley, N., 297, 310

369

370 INDEX

Bazley, N. and Fox, D.W., 297, 298, 303, 3l0(the Bazley and Fox base problem for bounds oneigen­values)

Beam equation, 326 Becker, R., 198, 366 Beltrami, E.J. , 198 Bending rigidity for thin plates, 230 Berkeley, bishop, 107 Bernoulli-Euler, beam hypothesis, 177, 217 Bernoulli, Daniel, 247, 248 Bernoulli, Jacob, 191, 242 Bernoulli, Johann (John) , 54, 70, 110 Bernoulli law, 177 Bernoulli's problem of "elastica", 247 Bessel's inequality, 362 Betti's theory, 15 Bickley, W. and Temple, G., 293, 308 Bing-Anderson theorem, 335 Birkhoff, G.D., 72, 106 Bliss, G.A., monograph on calculus of variations,

110, 191, 242 Blum, E.K., 198 Bochner, S., 364, 365, 454, 455 Bolza, 0., 95 Bolza's Problem, 96, 132, 133, 134, 135 Brachistochrone Problem, 54, 55, 100 Brachistochrone-Newton's solution, 55 Brachistochrone-type problems, 58 Browder, F., and Gupta, C.P.-decomposition of operators in Banach spaces, 118

Bruhat, Y.C., 198, 456 Bucciarelli, L.L., Dworski, Nancy, book on Sophie

Germain, 177, 192, 251 Buck, R.C., 202 Buckling-historical developments, 247, 308 Burton, T. and Grimmer, R., paper on non-continuability, 79, 106

Butkovskii, A.G., 199 Butkovskii, A.G., Egorov, A.I.i and Lurie, A.K.,

199 Butkovskii, A.G. and Lerner, A.Y., 198 Butkovskii, A.G. and Poltavskii, L.N., 198, 199

Calculus of variations(definition of), 57

INDEX

Canonical equations of Hamilton, 74 (also see Hamilton's canonical equations)

Caratheodory, C., 64, 106, 197 Caratheodory's accessibility theorem, 126 Caratheodory's variational form, 124 Cartan, E., 106, 200 Cartan's momentum energy tensor, 124 Cartesian coordinates, 121, 162, 206, 211, 213,

214, 240, 253, 298 Castigliano's theorem, 11, 12, 13, 14, 15, 222 Casti9liano's functional, 240 Cauchy, 352, 353, 355, 441, 442 Cauchy-Green strain tensor, 183 Cauchy-Schwartz inequality, 52, 350, 351, 353 Cauchy-Schwartz inequality-for beams(torsion and bending), 222

Cesari, L., 199 Cetaev, 142 Cheng, K.T, 226 Cheng and Olhoff paper, 226 Chladni, E.F., 176 Choi, K.C., 28 Ciarlet, P.G., 216, 244, 307 Ciarlet, P.G. and Destuynder, P., 216, 244 Closed graph theorem, 118 Closeness of order one, or higher, 60 Clough, R. and Wilson, E.L., 205 Coleman, B.D., 1 Coleman, B.D. and Gurtin, M., 199 Coleman, B.D., Markovitz, J., and Noll, W., 199 Collins, W., 191 (The) Commutator for the Lagrangian, 41 (The) complementary virtual work principle (equivalence with the existence of Gibbs' free energy), 159

Complete integral solution, 127 Conjugate convex functionals, 114 Conservation of energy, 72 Conserved Hamiltonian, 73 "Constitutive" variational principle, 208 Constraints for Fichera's energy inequalities,

222 Constraints-holonomic, 31,34 Constrained problems and Lagrangian multipliers,

82

371

372

Convex functionals, 114 Corben, H.C. and Stehle, P. monograph, 40, 104 Coupled bending and torsional vibrations of a

beam, 44 Courant, Richard, 103 Courant, R. and Hilbert, 367 Courant's N.Y.U. lectures on Dirichlet's principle, 103

INDEX

Critical point of a functional (definition) , 151 Critical point of a functional (discussion) , 152, 153, 154 Critical point - a connection with Legendre transformation, 154, 155, 191, 192, 193

Critical points-multiple, 155, 156, 157, 158, 159, 160, 161

Curvature of a shell surface, 237

d'Alambert, 3 d'Alambert's inertia forces, 3 d'Alambert's principle, 65, 66, 108 degrees of freedom, 29 Dielectric coefficients, 215 (The) "8" notation (a heuristic explanation), 31 (The) delta convergent sequences of Mikusi~ski, 139

den Hartog, J.P., 241, 254, 307, 366 Derivative-an abstract definition, 341 Derivative in a Hilbert space, 343 Derivative second-see second derivative Diaz, J.B., 298, 310 Dieudonne, J., 339 (The) Dirac bra-ket notation, 334 Dirac delta function, 46 Dirac equation, 327, 328, 332, 333 D'irac-Fock-Podolski, 130 Directional derivative, 95 Dirichlet, 189 Dirichlet's principle, 101, 102, 103, 104 Dirichlet's problem, 101, 121, 122, 131, 162, 303 Dini, 342 Djukic, D.S., 105 Do, Claude, 306, 312 (van) Doren, R., 144, 197 Double pendulum, 29, 30

INDEX

Drummond, J.E. and Downs, G.L. brachistochrone solution, 69, 105

dual Lagrangian integral, 41 dual Lagrangian for a harmonic oscillator, 42 duality between points and lines in Euclidean

geometry (relation to Legendre transformation) , 118, 119, 149

Duggan, R.C., 190, 200 Duhammel's linear law of thermoelasticity, 213 Duhammel's principle, 139, 230 Duhem's internal energy function, 189 Dunford, N. and Schwartz, J.T., 293, 308, 367 Duvant, G. and Lions, J.L. monograph, 197, 206,

243

Edmonds, J.D., 327, 328 Einstein, Albert, 107 Einstein's thought experiments, 107 Eigenvalues of the beam operator, 281, 283, 352 Eigenvalues of the plate operator, 285

373

Eigenvalues - dependence on parameters, 279 Eigenvalues-merging: see merging of eigenvalues Eigenvalues-sensitivity with respect to the design

parameters, 271 Eigenfrequencies for a vibrating plate-lower

bounds, 375, 376, 377, 378, 379 Eigenfrequencies-the base problem, 301 (The) energy balance in thermoelasticity, 213 Elasticity theory of Euler, 248, 249, 250 Elsgol'c L., 110, 191, 242, 311 Embedding of the stability of equilibrium problem in a larger space, 285

(The) energy approach to statistical equilibrium, 9, 10

(The) energy balance in thermoelasticity, 213 Energy product, 149 Energy product for beam in bending and torsion,

139 Engineering statics(use of finite element tech-nique), 216, 217, 218, 219, 270

Entropy introduced by Legendre transformation, 215 Equilibrium equation for thin plates, 229 Eringen, C., 188 Erdmann-Weierstrass corner condition, 62 Euclid's element's, 119

Euclidean plane, 118, 230, 294, 298 Euclidean measure, 230, 233, 412

INDEX

Euclidean distance approximation, 356, 357, 358 Euler, Leonhard, 241, 249, 250, 251, 252, 253,

254, 255, 261, 338 Euler1s angles, 331 Euler-Bernoulli beam, 136, 217 Euler1s buckling formula, 247, 248, 249, 250, 251,

252, 253, 259, 260, 261, 262 Euler-Bernoulli hypothesis in plate theory, 225 Euler1s critical load formula, 252, 253, 254, 255,

256, 257 Euler1s elastica theory, 250, 251, 252, 253 Euler-Lagrange, 189, 207 Euler-Lagrange equations, 39, 42, 57, 62, 63, 66,

67, 68, 69, 110, 111, 148, 170, 175, 181, 207, 208, 209, 214, 262

Euler-Lagrange, eq., an elementary deviation, 57 Euler-Lagrange, eq., for the dual Lagrangian, 41 Euler-Lagrange, non-existence of solutions, 63 Euler-Lagrange theory, 110 Euleri Leonhard~ Opera Omnia, 247, 248, 307 Euler1s, L. Opera Omnia, Truesdell1s comments,

110 Evans, D.J., 330, 331, 338 Eulerian strain, 158

Fermat1s principle of m1n1mum time, 55, 109 Finite element approach, 18 Fichera, G., 222, 24l, 297, 298, 310 (The) First law of thermodynamics, 182, 184 Flanders, H., 191 Flugge, w. 202, Flugge-Lotz, A contrib~tion to bang-bang prin-ciple, 134

Fluid flow - Sewell I s duality, 1).17, 178, 179 (The) Follower problem, 77 Forsyth, A.R., 110, 192, 242 Fourier coefficients, 289, 290 Fourier coefficients of the deflection function,

255, 256, 257 Fourier expansi<;m, 289 Fourier1s law of heat conduction, 213, 214 Fourier series, 113, 288, 289, 361, 362, 363, 364 Fox, J). W., 297, 310

INDEX 375

Frechet, M., 192, 242, 348 Frechet derivatives, 148, 149, 150, 151, 154, 161,

175, 220, 234, 235, 269, 271, 274, 275, 282, 313, 318, 319, 323, 324, 334, 335, 347

Frechet second derivatives, 161, 173 Frechet differentiation of a natural frequency,

8, 148, 149, 151, 179, 274, 315, 320, 322, 324, 333, 334, 335, 347

Friedman, Avner, 123, 198 Friedrichs, K.O., 111, 146, 192, 221, 243 Friedrichs' version of Legendre transformation,

221 Frobenius, G., 242 Functional - a definition, 57

G-the shear modulus (or of torsional rigidity) ,13 Gaida, R.P., 130, 199 Galerkin variational method, 293, 296 Gallagher, R.H., 205 Galileo's transformation, 130 Garding's inequality, 223, 224 Gateaux derivative, 149, 151, 154, 164, 322, 323, 345,346,347

Gateaux, R., 192, 238, 348 Gauss, F., 3 Gauss' principle of least curvature, 108 Gavurin, M.K., 192, 199 Geometric effects in buckling of rods, 262 Gel'fand, I.M., 192 Gel'fand, I.M., and Shilov, G. Ye., 174, 192 Gel'fand, I.M. and Vilenkin, N.Ya., 144, 192 Generalized coordinates, 31 . Generalized displacements, 28, 29 Generalized forces, 28,29 Geodesics on a sphere, 88 Germain, Sophie, 177 (See Lagrange-Germain theory) Germain's hypotheses, 182 Gibbs free energy, 215 Gibbs thermodynamics potential, 159 Global coordinates, 22 Goldstein, H., 104, 199, 366 Goldenveizer, A.L., 238, 244 Goldstein, H., 99 Gould, S.H., 298, 310

376

Goddard's problem, 134 Gradient of afunctional, 117, 148, 155 Gradient notation, 150 Gramm-Schmidt technique, 358, 359

INDEX

Green's function for th~ plate operator, 123,174 Green's theorem, 300, 301 Greenspan, D., 176, 203 Gupta, 118 Gurtin approach to se1f-adjointness, 182 Gurtin, MrE., 1, 203, 338, 339 Gurtin, M.E. internal variable theory, 188 Gurtin's treatment of constitutive properties,188 Guz, O.M., 187, 199 Guz, O.M., variational formulation, 187

Hadamard, 64, 189 Hadamard's well-posed criteria, 64 Hadamard's internal energy function, 189 Hahn-Banach theorem, 148 "the Hami1tonian",37-40, 110, 117, 127, 145, 153, ·172, 178, 215, 221, 234, 313, 314, 327,333, 336,

337 "Hamiltonian formalism", 154, 240 The Hamilton-Jacobi equation, 109, 126, 127, 128, 129, 130

Hamilton's canonical equations, 37-38, 73-74, 117, 144, 175, 221, 313, 327

Hamilton's canonical for a thin plate, 174-175 Hamilton's principle, 72, 110, 142, 236, 329 Hamilton, quaternionic algebra, 316 Hamilton's "theological" principle, 108 Hamilton, W.R., 108 Hardy, G.H., Littlewood, J.E. and Polya, G., 241,

308 Hardy, Littlewood and Po1va monograph, 258 Haug, E.J., 23 (The) Heat conduction tensor, 215 (The) H~at flux, 213 Hegel, H. 107 Hermann, Robert, 1 Herrera, I., and Bielak, J., 196, 200 Hertz, H., 25, 28, 105 Hertz's principle of least curvature, 74, 108 Hertz's principle of least trajectory, 108 Hessian determinant, 120

INDEX 377

Hilbert, 189 Hilbert's formulation of Dirichlet's principle,

103 125, 130 Hilbert's "independency integral",

Hilbert spaces, 111, 116, 118, 135, 144, 145, 146, 147, 149, 150, 152, 154, 155, 159, 161, 162, 163, 164, 165, 167, 173, 174, 281, 293, 294, 296, 298, 299, 304, 316, 321, 335, 336, 341, 342, 343, 344, 349, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362

Hilbert space H , M in plate theory, 233, 293, 358, 361, 362 ~ w

Hille, E. and Phillips, R., 364, 365 (the) Hodograph-Legendre transformation, 122,123 Holder, 226 Homogenization technique (see averaging) Hooke's law, 111, 121, 137, 146, 160, 186, 206,

225, 230, 262 Hooke's law for the thin plate theory, 225, 298 Hormander, L., 192, 364, 365 l'Hospital's rule, 265, 269 Hu's variational principle, 187 Hu-Washizu principle, 210, 240 Huygens, Christian, 8

Ilyushin, A.A., 190, 192 Ilyushin-Rivlin-Valanis theories, 190 (the) Incompatibility tensor, 159 (the) Inertia matrix, 18, 20 Infinitesimal displacement, 32 Internal variables, 189 Internal variables, Valanis' theory, 189, 203 Ivanov, N.A., 200

Jacobian, 157, 161, 343 Jacobi's theorem, 126 John, F., 105 Jordan curve, 230

Kant, Emmanuel, 107 (the) Kantorovich variational method, 297 Kantorovich, L.V. and Akilov, G.P., 193, 243, 364,

365 Kantorovich, L.V. and Krylov, V.L., 309 von Karman, 306

378 INDEX

(T. von) Karman and Biot, M.A., 106 Kato, T.,117, 118, 144, 153, 193, 243, 293, 297,

298, 308, 310 (the) Kinematic metric in rigid body mechanics, 329

Kinetic energy terms, 17 Kinetic energy for a rigid body, 17, 18, 21 Kinetic energy of a vibrating string, 6 Kirchhoff's energy approach to stability, 254,

255, 259 Kirchhoff's law, 255, 261 Kirchhoff-Love theory of shells, 240, 241 Kirchhoff's shell theory hypothesis, 241 Kirchhoff's hypothesis, 181, 241 Kitahara, M., 309 Kleinschmidt, von W. and Schulze, H.K., brachistochrone problem in a central force field, 70, 105

Klein-Gordon operator, 326 Kneser, A., 106, 126, 204 Kohn, R. and Vogelius, M., plate estimates, 227,

229, 244 Kolomy, J., 193 Komkov, Vadim, 106, 193, 194, 198, 200, 243, 308,

309, 339 Komkov's criterion for non-continuability of solutions, 79

Komkov's embedding technique for the buckling problem, 284, 285-296

Komkov and Valanis' hidden variable approach, 40 Korn, A., Ill, 146, 197, 227, 229 Korn's duality, 144 Korn's inequality, 227, 229 Kosmodemianskii, A.S. and Lozhkin, V.N., 244 Kuroda, S.T., 298, 311 Kuti, I., 196

Lagrange, J.L., 3, 31, 109, 177, 247, 248, 250, 254, 260, 307, 326

Lagrange Germaine Derivation, 180 Lagrange-Germaine equation, 115, 169 Lagrange-Germaine plate formula, 169, 177, 180 Lagrange-Hamilton,144, 146 Lagrange's bilinear form, 224 (the Lagrangian,37, 110, 117, 313, 314, 315, 316,

INDEX

327, 328, 337, 338 Lagrangian formalism, 154, 188 Lagrange equality of mean values of potential

and kinetic energy, 235 Lagrangian, constrained, 86, 123 Lagrangian functional, 155, 173, 179, 234, 337 (the) Lagrangian for a two-body problem, 43, 44 Lagrangian density function, 108

379

Lagrangian (integral) functional, 155, 315 Lagrangian, modified (non-constrained), 87 Lagrangian multipliers,85, 86, 135, 142, 220, 276,

301 Lagrangian multipliers, an elementary example,

81, 82, 83, 84, 85, 86, 87 Lagrangian product, 182 Lagrange's equality of mean values of potential

and kinetic energy, 235 Lakshmikantham, V. and Lee1a, S., 223, 241 Laminated (multilayer beams), 217 Lamme parameters, 237 Lamme coefficients, 211 Lanczos, G., 110, 196 Landau, L.D., 30 Landau, Lev. D. and Lifshitz, E.M., 104, 110,

127, 197, 367 Langhaar, H.L., 110, 197 Laplace equation, 115, 121, 169, 325 Laplace operator-quaternionic representation, 297,

302, 325 Lattman, 330 Lax-Milgram theorem, 164, 321, 322, 324, 361 Lax-Mi1gram-quaternionic version, 325 Lebesgue, 189 Legendre, 177 Legendre duality, 172 Legendre's strong condition, 99 Legendre's test, 99 Legendre transformation,lll, 118, 119, 120, 121,

125, 132, 144, 145, 154, 155, 159, 167, 171, 173, 181, 215, 219, 220, 221, 319, 331

Legendre transformation-geometric interpretation, 118, 119, 120, 121

Legendre transformation for the membrane problem, 121, 122, 123

Legendre-hodograph, see hodograph transformation

380 INDEX

Legendre transformations of a higher order, 123 Leonhardi Euleri Opera Omnia, 191, 247, 307 Levi, 223, 257 Lewi, Hans, 315 Lichnerowicz, A., 1 l'Hospital's rule(application to Rayleigh's quotient), 269

Lie algebra, 2 Lie groups, 2 Lifshitz, E.M., 25 Lin's variational principle, 179 Lions and Magenes, 206 Lorentz transformation, 130 Love, A.E.H., 200, 244 Lubliner, J., 188, 200, 201 Lumped parameters, 18 L - sensitivity of the operator L with respect u to a vector u, 281, 282

M-m Castigliano-Betti formula, 16 Mach, Ernst, 25, 105 Mach's cosmological philosophy, 107 MacLane, S., 197 Magenes, J., 206 Magnetic manopole current, 328 Mansfield, E.H., 200 Mariotte and Leibniz, 253 Masur, E. F., 226 Material indifference principle, 182 Material (intrinsic) time scale, 184 de Maupertuis, M. principle (derived from Legendre's transformations), 123, 249

de Maupertuis, M., principle of least action, 71, 109

(the) Mass matrix, 21 Maxwell, Clark, 328, 329 Maxwell-Morrera stress functions, 150, 158 Maxwell equations, 326, 328, 329 Membrane deflection(the soap bubble problem), 112,

113 Merging of eigenvalues(singular design point), 276,

279 Miersemann, E. 282, 306, 311

INDEX

Mikhlin,(Holt, Rinehart, and Winston), 366 Mikhlin, S.G., 194, 201, 243, 367 Mikusinski, J.,139, 185, 201

381

Mikusinski's definition of convolution operators, 185

Milgram-Lax theorem(see Lax-Milgram), 164 Mindlin's free energy, 215 (the) Minimum potential energy principle, 9 (the) von Mises catastrophe, 28, 29 Mitrinovic, D.S. (monograph on analytic inequal-ities), 223, 241, 258, 307

Moment of inertia matrix, 18 Moment of inertia of cross-section I(x), 217,254 Momentum energy tensor of Cartan, 124 Moreau, J.J., 179, 203 Morley, L.S.D., 201, 202 Morse, P.M. and Feshback, H., 110, 201, 366 Multiply-layered laminated beams, 217 Mushtari, Kh.M., 245 (the) Mushtari-Galimov book, 241, 245 Muskhelishvili, 1'1.1., 161, 201, 366

Nashed, M.Z., 194, 202, 333, 339, 348 Nastran Code, 19 Natural frequencies for a vibrating string,6,7 Natural frequencies for a plate (Weinstein's

approach to lower bounds), 297-304 Natural modes, 7 Navier, 177 Negative gravity for a pendulum, 81 Negative resistance, 80, 81 Nemat-Nasser, S., 197 Nashed, M.Z., 194, 202 Neumann, J. V., 144, 194, 243 (the) Neumann problem, 121, 131 Newton, Isaac, 3, 8, 70, 71, 105 Newtonian inertia forces, 3 Newton's second law, 3, 78, 214 Newton's second law for thermoelastic equilibrium,

214 Newton's solution of the brachistochrone problem,

109 Nikolai, E.L., 241, 254, 307 Noble, Ben, 40, Ill, 146, 153, 194, 331, 333, 338 Noble's two-sided inequalities, 328

382

Noble, B. and Sewell, M.J., 202 Noble's Hamiltonian systems, 117, 118

INDEX

Noble's duality for thermoelastic materials, 182, 183, 184, 185, 186, 187, 188

Noll, Walter, 1 Non-continuability of solutions in Euler-Lagrange equations, 62

Non-existence of optimal column design, 272 Non-positive mass, 81 Nonlinear analogue of Rayleigh's theorem, 273 "Normality" constraint for column buckling, 264 Novozhilov, V.V., 156, 244 Novozhilov and Finkelstein, 236, 240, 245 (the) Novozhilov matrix, 238 Novozhilov's monograph on thin shells, 240

Oden, J.T., 205, 243 Oden, J.T. and Reddy, J.N., 110, 194, 206, 243 Olhoff, N. and Rasmussen, S.H., 226, 276, 308 Optimal control of a vibrating beam, 136 Optimal design problem for a column, 275, 276,

277, 278, 279 Optimal shape-nonexistence of smooth solution,

277

Panovko, Ya.G. and Gubanova, 1.1., 311, 241, 254, 307

Parseval's equality, 359 Pauli matrices, 332 Payne, L.E., 105, 310 Pendulum elastic, 81 Perron's counterexample, 103 Pfafian form for the Hamiltonian, 73 Piezoelastic constants, 215 Piezoelastic phenomena, 215, 216 Piezoelasticity, lBO, 215 Planck, M., 110 Planck's critique of the ultra-violet catastrophe,

110 Planetary gear systems, 89 Plate geometry, 225, 226 Poincare, Henri, 72, 106, 223 Poincare inequality, 229, 257, 258, 259 Po~ncare integral invariant, 124 POl.sson, 177

INDEX

Poisson equation, 112 Poisson's ratio v - constraints on its value,

176, 180, 211, 231, 298

383

Polar moment of inertia of a beam with respect to the shear center, 45

Polyak, L.S., 196 Pontryagin, L.S., 97, 135, 167, 197 Pontryagin's constrained problem, 135, 142 Pontryagin's Hamiltonian, 97, 136, 141, 143, 144 Pontryagin's principle, 132, 133, 136, 140, 142,

144 Pontryagin's variational problem, 143 Pontryagin's problem, 96, 143 Popov, I.P., 312 Potential energy of a deflected string, 6 Potential energy of a double pendulum, 29 (the) Prager-Schields mutual compliance inequality, 222

Prager, W., 202, 366 (the) Prigogine-Glansdorff functional, 214 (the Principle of virtual work, 35, 36

Quaternions, 317-338 Quaternion algebra, 317 Quaternion valued functions, 321 Quaternion valued left and right derivatives, 318 Quaternion Legendre transformation, 319 Quaternion Lagrangian, 319

Quaternion symmetric derivatives, 321 Quaternionic version of Riesz representation

theorem, 321, 322 Quaternionic representation theorem, 324 Quantum mechanics via quaternion algebra, 327

Rabczuk, R., 223, 241 Rall, L.B., 117, 153, 194, 202 Ramm, A.G., 304, 306, 312 Raviart, P.A. and Thomas, J.M., monograph, 205 Rayleigh, Lord, 3, 25, 104, 257, 285, 308 Rayleigh and Ritz techniques, 290,293,296 Rayleigh's principle, 7, 257 (the) Rayleigh quotient, 8, 169, 265, 266, 269, 272, 273, 287, 290, 294, 296, 297, 304

Rayleigh's theorem, 274 Rebiere, J.P. and Sahraoui, S., 307

384

Reiss, E.L., 308 Reissner, 202, 221 (the) Reissner-Hellinger principles, 208 Rellich's inequality, 225 Riccati equations, 250, 251 Ricci equations, 150, 159 Riemann integral, 343

INDEX

Riesz representation theorem,146~148,163,164,360 Riesz, F. and Nagy, Sz., 367 Rigged Hilbert space, 174 Rigidity of solutions to variational problems,64 Ritz approximation, 296 Ritz, W., 287, 290 293 308 Ritz's direct vari~tion~l technique, 293 Ritz-Rayleigh and Galerkin variational methods, 293 Rivlin, R.S., 184, 194 Robinson, J., 205 Robinson, P.D., 190, 194 Rothe, E.H., 195 Routh, E.J., 25, 105 Rund, H., \ 204 Russalovs~aya, A.V., Ivanov, G.I. and Ivanov, A.I., solution of brachistochrone problem with friction, 70, 105

Sandhu, R.S. and Pister, K.S., 201 Sanso, F., 330, 331, 339 Sanso, F. and Evans, D.J., Kinematic coordinates,

330 Schwartz, Laurent, 189 Schrodinger equation, 332 Schulze, H.K. (See Kleinschmidt) Second derivatives regarded as tensor products,

167-169 (the) Second law of thermodynamics, 183 Second Frechet derivatives, 166, 313 Segel, L.A., 189, 201 Seide, Paul, 244 Semitses, G.J., 226 (the) Sensitivity of the natural frequency of a vibrating string, 7

Sensitivity of the column eigenvalues, 271, 281, Sensitivity of the natural frequency of a vibrating beam, 282

INDEX

Sensitivity of a vibrating plate, 282 Sewell, M.J., 178, 179, 190, 195, 196, 198, 202 Shell theory, 236, 237, 238, 239, 240, 241 Small deflection theory (almost quadratic energy

form), 263 Smoothness of solutions of elliptic equations,

225 Sobolev, S.L., 195, 300, 367

385

Sobolev class of functions, 131, 233 (the) Sobolev embedding lemma, 287 Sobolev's generalized derivatives, 300 Sobolev norm, 61 180, 287, 294 Sobolev spaces HI(n), w~'2(n), 113, 147, 180j 183,

218, 219 Sobolev spaces, H~[O,l], 261, 287 Sobolev spaces, Hri[O,l], 274 Sokolnikoff, I.S., 203 Splitting of operators, 172 Stampacchia, 206 State equations for piezoelastic materials, 215 State of a mechanical system, 139 Static bending of beams, 217 Stationary behavior of the critical column load

(non-linear theory), 268, 276, 277 (the) Stiffness matrix, 21, 22, 23 Strang, G. and Fix, G.J. monograph, 205 Strain energy of a bar in torsion, 11 Strain energy of a curved rod in bending, 13,14 Strain energy of a structure, 12, 20, 23 Strong minimization, 39 Synge, J.L., 190, 198 Snell's law of optics, 109 Szarski, J., 223, 241

Taylor, A.E., 307 Temple and Bickley, 293 Tensor products, 162, 163 Thermoelastic principles, 213, 214, 215 Thermoelastic coefficients, 215 Tiersten, H.F.,181, 182 Tikhonov, A.N. and Samarskii, A.A., 366 Timoshenko, Stephen, 39, 131, 203, 242, 311, 314, 255, 284, 308

Timoshenko's example of instability, 284 Tirnoshenko, S. and Gere, J.M., 242, 255,307

386

Timoshenko, S. and Goodier, J.N., 366 Tonelli's theorem{lack of) for quaternionic derivatives, 324

Tonti, E., 196 (the) Topological dual, 28 (the) Total derivative, 214 Torsional flywheel system, 39 (the) Transversal manifold, 125 Transversality, transversal curve, 125 Trefftz, 297

INDEX

Truesdell, Clifford, 1,200, 204, 242, 247, 249, 250, 254

Truesdell and Bharata monograph, 188, 204 Truesdell C. and Noll, W., 195 Truesdell, C. and Toupin, R., 195

Uniqueness of T*T decomposition for a positive operator, 116, 118

Universal variational principle, 187

Vainberg, M.M., 195, 196, 203, 243, 333, 335, 339, 348

Vainberg, M.M. and Kacurovski, R.R., 195 Vainberg lemma, 167 Vainberg theorem, 165, 328 Valanis, K., 188, 189, 195, 203 Valanis, K.C. and Komkov, V., 40, 106 van der Waerden equation, 332, 333 Vasil'ev, A.N. and Kazanskii, A.K. (article on hiqher order Legendre transformations), 123, 196

Vekovischeva, I.A., 181, 200, 244 Vibrating string, 3, 8 Vibrating Euler-Bernoulli beam, 136 Vibration of a circular plate, 283 Virtual displacements, 32 de Veubeke, E. Fraeijs, 156, 196, 205 von Kleinschmidt, W. and Schulze, H.K., (see

Kleinschmidt, von W.) von Mises catastrophe, 25

Washizu, K., 196, 240 Wave equation, 6 Weierstrass, K., collected works {Meier und Muller Verlag), 106

INDEX 387

Weierstrass analysis of Dirichlet's problem and Dirichlet's principle, 102, 103

Weierstrass approach to calculus of variations, 96

(the) Weierstrass E-function,93,94,98 Weierstrass, llO, 189 Weinberger, H., 293, 309 Weinstein, A., 297, 298, 303, 310, 311, 379 Weinstein, Fichera, Bazley, Fox, and Kato theory,

297 Weinstein's determinant, 298, 303 wirtinger's inequality, (also called Wirtinger­

Poincare-Almansi inequality), 223, 257, 258, 259

Yosida, K., 308, 364, 365, 367 Young, L.C., 189, 203 Young-Fenchel transformation, 130, 131, 132 Young-Fenchel duality, 131 Young's modulus, 12, 46, 180, 211, 247, 250, 253,

283, 298, 302, 316 Yourgrau, W. and Mandelstam, S., 110, 204

Zienkiewicz, O.C., 204 Zorn, M., 196