|
|
 |
| Àâòîðèçàöèÿ |
|
|
|
| Ïîèñê ïî óêàçàòåëÿì |
|
|
|
|
|
|
|
 |
|
|
|
| Weinstock R. — Calculus of variations with applications to physics & engineering |
|
|
|
| Ïðåäìåòíûé óêàçàòåëü |
Abraham, M. 294n. 316
action 85—88 268—269
Approximation of eigenvalues see Ritz method
Area, formula for surface 10
Area, maximum, in plane 53—56
Area, minimum surface 115
Area, minimum surface of revolution 19—20 30—32
Asymptotic distribution of eigenvalues see Eigenvalue-eigenfunction problems
Bar bending, by couples 212—217 250—251
Bar bending, “engineering theory” 251. See also Eigenvalue-eigenfunction problems; Vibrations
Basic lemma 16—17 43—44
Becker, R. 294n. 316
Bending by couples, bar 212—217 250—251
Bending by couples, plate 224—228
Bending moment 214—215 250
Bernoulli — Euler equation 251
Bernoulli, J. 19 29 67 71
Bessel functions 129—130
Bessel functions in circular-membrane problem 194
Bessel functions in circular-plate problem 259
Bibliography 319
Black-body radiation 198 271
Bliss, G.A. vii 2n. 20n. 29n. 31n. 319
Body forces 199 205
Bohr, N. 264 275
Bolza, O. vii 2n. 319
Born, M. 277n.
Boundary conditions, distinction between free and imposed 114—115
Boundary conditions, elastic solid, general 209—210
Boundary conditions, electrostatic potential 294—295
Boundary conditions, relaxed conditions 302—306
Boundary conditions, vibrating bar 221
Boundary conditions, vibrating membrane, elastically held edge 152
Boundary conditions, vibrating membrane, free edge 153
Boundary conditions, vibrating plate 231—236 256—257
Boundary conditions, vibrating plate, free-edge square 248—249
Boundary conditions, vibrating plate, free-edge square, corner condition 249
Brachistochrone 19 28—29
Brachistochrone, between two curves 47
Brachistochrone, curve to fixed point 41—43
Brachistochrone, optical solution 71
Brachistochrone, point, to curve 40—41
Brachistochrone, point, to vertical line 38
Capacity of condenser 297
Capacity of condenser, approximation of, from above 298—300 314—315
Capacity of condenser, approximation of, from above, cubical condenser 315
Capacity of condenser, approximation of, from above, ellipsoidal condenser 300—302 316
Capacity of condenser, characterization, maximum 304
Capacity of condenser, characterization, minimum 297
Capacity of condenser, cylindrical 309
Capacity of condenser, cylindrical, approximation of 317—318
Capacity of condenser, cylindrical, characterization, maximum 309—310 317
Capacity of condenser, cylindrical, characterization, minimum 309 310
Capacity of condenser, from below 302—306 316
Capacity of condenser, from below, cubical condenser 316—317
Capacity of condenser, from below, ellipsoidal condenser 306—308
Cavity radiation 198 271
Cavity radiation, asymptotic distribution of frequencies 198
Change of variables see Transformation
Compatibility, equations of 204
Condenser see Capacity of condenser
Coordinates, generalized 73
Couple see Bending by couples
Courant, R. viii In. 171n. 311n. 319
Davisson, C.J. 264 270
de Broglie, L. 264 270 271
Degeneracy 154. See also Eigenvalue-eigenfunction problems
Derivative of an integral 5
Determinants 8—9
Dirichlet, existence of minima of integral 310—311
Dirichlet, principle of 311
Dirichlet, problem of 296
Dynamics of particles 72—92
Eigenvalue-eigenfunction problems 99
Eigenvalue-eigenfunction problems, approximation 107—114 117—118
Eigenvalue-eigenfunction problems, asymptotic distribution 196—197
Eigenvalue-eigenfunction problems, characterization, isoperimetric 116
Eigenvalue-eigenfunction problems, characterization, minimum 102—105
Eigenvalue-eigenfunction problems, characterization, normalization 99 101
Eigenvalue-eigenfunction problems, characterization, orthogonality 101
Eigenvalue-eigenfunction problems, quantum mechanics 263
Eigenvalue-eigenfunction problems, quantum mechanics, characterization, isoperimetric 263 280
Eigenvalue-eigenfunction problems, quantum mechanics, characterization, minimum 281
Eigenvalue-eigenfunction problems, quantum mechanics, eigenvalues as energy levels 263 278
Eigenvalue-eigenfunction problems, quantum mechanics, significance of normalization 276 279.
Eigenvalue-eigenfunction problems, Sturm — Liouville 119—131
Eigenvalue-eigenfunction problems, Sturm — Liouville, asymptotic distribution 196—197
Eigenvalue-eigenfunction problems, Sturm — Liouville, characterization, isoperimetric 119
Eigenvalue-eigenfunction problems, Sturm — Liouville, maximum-minimum 196
Eigenvalue-eigenfunction problems, Sturm — Liouville, minimum 130
Eigenvalue-eigenfunction problems, Sturm — Liouville, orthogonality 130. See also Sturm — Liouville problem
Eigenvalue-eigenfunction problems, three-dimensional vibrations 197—198
Eigenvalue-eigenfunction problems, vibrating bar 221—224
Eigenvalue-eigenfunction problems, vibrating bar, clamped 244—246
Eigenvalue-eigenfunction problems, vibrating bar, explicit solution 252—255
Eigenvalue-eigenfunction problems, vibrating bar, isoperimetric characterization 223
Eigenvalue-eigenfunction problems, vibrating bar, orthogonality 223—224
Eigenvalue-eigenfunction problems, vibrating membrane 145—146 152
Eigenvalue-eigenfunction problems, vibrating membrane, approximation 188—192
Eigenvalue-eigenfunction problems, vibrating membrane, asymptotic distribution 177—188
Eigenvalue-eigenfunction problems, vibrating membrane, characterization, isoperimetric 147—148 152—153
Eigenvalue-eigenfunction problems, vibrating membrane, degeneracy 154 155 163—164 167n.
Eigenvalue-eigenfunction problems, vibrating membrane, maximum-minimum 171—173
Eigenvalue-eigenfunction problems, vibrating membrane, maximum-minimum, consequences 173—176
Eigenvalue-eigenfunction problems, vibrating membrane, minimum 164—167
Eigenvalue-eigenfunction problems, vibrating membrane, minimum, consequences 167—170
Eigenvalue-eigenfunction problems, vibrating membrane, minimum, limitations 170—171
Eigenvalue-eigenfunction problems, vibrating membrane, normalization 146 152
Eigenvalue-eigenfunction problems, vibrating membrane, orthogonality 154—157
Eigenvalue-eigenfunction problems, vibrating membrane, rectangular 161—164
Eigenvalue-eigenfunction problems, vibrating plate 236—240
Eigenvalue-eigenfunction problems, vibrating plate, approximation 240—244
Eigenvalue-eigenfunction problems, vibrating plate, characterization, isoperimetric 237—238 257
Eigenvalue-eigenfunction problems, vibrating plate, characterization, maximum-minimum 240 258
Eigenvalue-eigenfunction problems, vibrating plate, characterization, minimum 239—240 258
Eigenvalue-eigenfunction problems, vibrating plate, circular 259—260
Eigenvalue-eigenfunction problems, vibrating plate, clamped square 244—248 260
Eigenvalue-eigenfunction problems, vibrating plate, degeneracy 237
Eigenvalue-eigenfunction problems, vibrating plate, orthogonality 238—239
Eigenvalue-eigenfunction problems, vibrating string 98—100 115—116
Einstein, A. 271
Elasticity theory 199—260
Elasticity theory, approach to dynamical problems 210—211
Elasticity theory, boundary conditions, general 208—210
Elasticity theory, equations, of equilibrium and motion, general 209
Electrostatic intensity 294
Electrostatics 294—318
Electrostatics, two-dimensional problems 308—310
End-point conditions see Boundary conditions; Undetermined end points
Energy, total 75—76. See also Kinetic energy; Potential energy
Energy, total, eigenvalues in quantum mechanics 263 278
Energy, total, eigenvalues in quantum mechanics, helium atom 286
Energy, total, eigenvalues in quantum mechanics, one-electron atom 275
Energy, total, equivalence with hamiltonian 77
Energy, total, equivalence with hamiltonian, in least-action principle 85—88
Energy, total, equivalence with hamiltonian, in reduced Hamilton — Jacobi equation 83
Equations of compatibility 204
Equilibrium 90
Equilibrium of elastic plate 260
Euler — Lagrange equations 20—23. See also Transformation
Euler — Lagrange equations, degenerate cases 25—26 134—135
Euler — Lagrange equations, first integrals 24—25 34 45 52
Euler — Lagrange equations, higher derivatives 46 61 64
Euler — Lagrange equations, in dynamics of particles 75 92
Euler — Lagrange equations, isoperimetric problems 50 95 134
Euler — Lagrange equations, several dependent variables 32—33 52 60
Euler — Lagrange equations, two or more independent variables 94 95 133 134
Euler's theorem on homogeneous functions 6
Extremizing function 23
Extremum 2 23
| Extremum of double integral 93—95
Extremum of n-tuple integral 133
Extremum of triple integral 132—133
Fermat's principle 67—71
Fermat's principle in quantum mechanics 268—269
Flexural rigidity, of bar 217
Flexural rigidity, of plate 228
Force components, generalized 89—90
Fox, C. 319
Franklin, P. 3n. 319
Fundamental lemma 16—17 43—44
Generalized coordinates 73
Generalized force components 89—90
Generalized momenta 76
Generalized velocity components 74
Geodesies 18 26—27 61—63 65
Geodesies in plane 17—18 24
Geodesies on cone and cylinder 45
Geodesies on sphere 27—28 62—63
Geodesies on surface of revolution 28 45
Germer, L.H. 264 270
Goldschmidt discontinuous solution 31
Goursat, E. 3n. 319
Gradient 12
Green's theorem 12—15
Group velocity 290
Hamilton equations of motion 78
Hamilton equations of motion, transformed 79—82
Hamilton integral 74
Hamilton — Jacobi equation 82
Hamilton — Jacobi equation, in quantum mechanics 262 269 277
Hamilton — Jacobi equation, reduced 83 84
Hamilton's principle 74—75 77
Hamilton's principle, application to vibrations, of bar 218—221
Hamilton's principle, application to vibrations, of membrane 145 148—151 159
Hamilton's principle, application to vibrations, of plate 229—231
Hamilton's principle, application to vibrations, of string 96—97 105
Hamilton's principle, extended 88—90 92
Hamilton's principle, extended, application to motion, of elastic solid 205—209
Hamilton's principle, extended, application to motion, of membrane under transverse force 193
Hamilton's principle, extended, application to motion, of plate under transverse force 260
Hamilton's principle, extended, application to motion, of string under transverse force 116
Hamiltonian 76—77 79 82
Hartree model of the atom 286—290
Hedrick, E.R. 319
Helium atom 281
Helium atom, approximation of ground-state energy 281—286
Helium atom, singly ionized 275
Hermite differential equation 131
Hilbert, D. viii 171n. 311 319
Homogeneous functions, Euler's theorem on 6
Hooke's law 203—204
Hydrogen atom 271—275
Hydrogen atom, energy levels 275
Hydrogen atom, spherically symmetric wave functions 274
Ince, E.L. 99n. 121n. 124n. 125n. 319
Indeterminacy, principle of 277
Integration by parts 5
Isoperimetric problems 48—57 65.
Isoperimetric problems, two or more independent variables 95 133—134
Jackson, D. 127n. 319
Jacobi, C.G.J. vii. See also Hamilton-Jacobi equation
Jacobian 9—10 136 140—141 293
Kellog, O.D. 3n. 296n. 311n. 319
Kent, G. viii
kinetic energy 73
Kinetic energy of elastic deformation 205
Kinetic energy of vibrating bar 218
Kinetic energy of vibrating membrane 143
Kinetic energy of vibrating plate 228
Kinetic energy of vibrating string 96
Kronecker delta 101
Lagrange equations of motion 75 92.
Lagrange multipliers 6 49 63.
Lagrangian 74
Laguerre polynomials 128—129 274
Laplace's equation 295 309
Laplacian 12
Laplacian, transformation of 138—142
Least action, principle of 85—88 268—269
Legendre differential equation 131
Legendre, A.M. vii
Lemma, basic 16—17 43—44
Line integral 6 7
Linear independence 9
Love, A.E.H. 319
Maximum-minimum characterization of eigenvalues see Eigenvalue-eigenfunction problems
Mechanics see Dynamics of particles; Quantum mechanics
Membrane 142. See also Eigenvalue-eigenfunction problems; Vibrations
Membrane system, definition of 167
Membrane system, “narrower” relationship 168
Minimum characterization of eigenvalues see Eigenvalue-eigenfunction problems
Moment, bending 214—215. 250
Moment, of inertia (area) 214
Momenta, generalized 76
Morse, P.M. 259n.
Newton, I. 19 74 75
Normal derivative 7—8 11—12
Normalization 99. See also Eigenvalue-eigenfunction problems
Optics, geometric see Fermat's principle
Orthogonality 101. See also Eigenvalue-eigenfunction problems; Schmidt process of orthogonalization
Parametric representation 34—36
phase velocity 266 267 269
Phase velocity of a particle 269
Piecewise continuity 4
Piecewise differentiability 4
Planck's constant 262n. 271
Planck, M. 271
Plate bending, by couples 224—228
Plate bending, by transverse-force distribution 260. See also Eigenvalue-eigenfunction problems; Vibrations
Poisson's ratio 203—204
potential energy 72 89
Potential energy of elastic deformation 205 212
Potential energy of elastic deformation, bar bent by couples 216—217
Potential energy of elastic deformation, plate bent by couples 227—228
Potential energy of elastic deformation, vibrating bar 218
Potential energy of elastic deformation, vibrating plate 228
Potential energy of electrostatic field 295
Potential energy of helium atom 281
Potential energy of many-electron atom 287
Potential energy of one-electron atom 272
Potential energy of vibrating membrane 143—145
Potential energy of vibrating membrane, elastic binding of edge 148
Potential energy of vibrating string 96
Potential, electrostatic 294
Potential, minimum characterization 295 311—312
quantum mechanics 261—293
Quantum mechanics, comparison with classical mechanics 276—277
Quantum mechanics, reduced Hamilton — Jacobi equation in 262 269 277.
Radiation 198 271
Rayleigh — Ritz method 241ra. See Ritz method
Rayleigh, Lord, J.W.S. 247n. 249n. 319
Ritz method 241
Ritz method for helium atom 281—286
Ritz method for vibrating membrane 188—192
Ritz method for vibrating plate 240—244
Ritz method for vibrating plate, clamped square 244—248
Ritz method for vibrating string 107—114
Ritz method in electrostatics 296
Ritz method in Hartree method 286—290
Ritz, W. 241
Rod see Bar bending
Rope, hanging 56—57 64 66
Saint — Venant's principle 211n.
Scalar product 12
Schiff, L.I. 319
Schmidt process of orthogonalization 155—157
Schroedinger equation, first derivation 262—263
Schroedinger equation, for several particles 277—280
Schroedinger equation, for single particle 263 270
|
|
|
| Ðåêëàìà |
|
|
|
|
|
|
|
|
Î ïðîåêòå