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Eisenhart L.P. Ч Continuous groups of transformations
Eisenhart L.P. Ч Continuous groups of transformations

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Ќазвание: Continuous groups of transformations

јвтор: Eisenhart L.P.

јннотаци€:

This book sets forth the general theory of Lie and his contemporaries and the results of recent investigations with the aid of the methods of the tensor calculus and concepts of the new differential geometry. The first three chapters contain in the main the results of the first period. Chapter Four is devoted to the theory of the adjoint group and the sequence of theorems basic to the characterization of semi-simple groups, as developed by Cartan and recently by Weyl and Schouten. Although geometrical ideas are used throughout the book, Chapter Five contains the geometrical applications of the theory both in the space of the transformations and in the group-space, and here are to be found particularly the concepts of the new differential geometry. The theory of contact transformations with applications to geometry and mechanics is set forth in the closing chapter.


язык: en

–убрика: ћатематика/—имметри€ и группы/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1961

 оличество страниц: 159

ƒобавлена в каталог: 29.04.2005

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
$A_{\alpha}^{a}$, $A_{\alpha}^{a}$, $\bar{A}_{\alpha}^{a}$, $\bar{A}^{\alpha}_{a}$      20Ч23 27Ч30 32 161 198Ч201 206 206 229
$T_{\alpha}$      14
$U_{\alpha}^{a}$, $U^{\alpha}_{a}$, $\bar{U}_{\alpha}^{a}$, $\bar{U}^{\alpha}_{a}$      51Ч55
Abelian group      48Ч50 86 88 113 137 166 160 162 183 184 192 218Ч220 227
Absolute, invariant of a group      61 62 67 68 93 96 96 107
Adjoint group      151Ч166 160Ч163 170 183 199
Affine group      43 67 66
Affine group, parameter      199
Amaldi      267 296
ARC      199 212
Asymmetric connection      197 198 230 234
Asystatic group      84 86 123
Automorphism, $B^{h}_{ijk}$      196Ч198 232 233
Automorphism, $B_{\beta\gamma}$      206
Automorphism, of a group      151
Automorphism, of a space      230 231 236
B$\hat{o}$her      69 146 177 294
Base, vector      141
Base, vector, change of      141 142
Basis of a function group      282 285 288
Basis, change of      28
Basis, of a group      28
Berwald      277 295
Bianchi      26 44 66 66 71 82 86 108 121 123 133 137 138 212 216 218 220 228 229 236 293 294
Bianchi, identity of      232
BOM      280 296
Bortolotti      237 296
Burnside      183 294
Burnside, $C^{e}_{ab}$      22 26 27
Burnside, $C_{abe}$      162 206
Canonical form of a function- group      286 290
Canonical, form of transformation      145Ч160
Canonical, parameter      46 47 62 160 161 160
Cartesian coordinates      188
Carton      66 167 169 173 174 179 180 182 184 199 201Ч208 213 236 237 293Ч296
Characteristic, equation      140Ч142 165Ч160 164 169 172 177-179
Characteristic, function      262 260 262 274 278 280 292
Characteristic, matrix      166
Christoffel symbol      188 236
Classification of groups      166 180Ч182
Clifford, parallelism of      237
Coefficient of linear connection      47 192
Cohen      91 294
Comitant      161
Commutative,      113
Commutative, function-groups      283 287 288 290
Commutative, functions      283
Commutative, groups      112 114Ч117 121 122 256
Commutative, transformations      110 112
Commutator      6
Complete system of differential equations      7 8 12 86
Completely integrable differential equations      1 4
Complex, linear      186
Composition, constants of      22
Composition, indices of      128
Composition, series of      127Ч134
Conformal spaces      236
Connection, $(+)-$      199 230
Connection, $(-)-$      199 230
Connection, (0)$-$      199 206 231
Connection, asymmetric      197 198 230 234
Connection, coefficients of      47 192
Connection, linear      45 47 192
Connection, symmetric      194 198 233 234 236
Constant, of composition      22
Constant, of composition of structure      22 26 28 43 48 56 112 126 151
Contact transformation, geometrical properties      242Ч246 267 267Ч277
Contact transformation, geometrical properties of maximum rank      263Ч273 276
Contact transformation, geometrical properties, homogeneous      239Ч266 263Ч278 291 292
Contact transformation, geometrical properties, infinitesimal      251Ч266 260 262 281 292
Contact transformation, geometrical properties, non- homogeneous      255Ч263 281
Contact transformation, geometrical properties, restricted non-homogeneous      260Ч263 280 281 286 287 291
Continuous group, finite      16 17
Continuous group, finite, infinite      15
Continuous group, finite, mixed      68
Contravariant, tensor      187
Contravariant, vector      26 36 187
Cosine of angle      209 212
Covariant, derivative      189
Covariant, differentiation      189 196 198 232 233
Covariant, normal      239 242Ч244 246
Covariant, tensor      187
Covariant, vector      27
Curvature,      195Ч198 206 207
Curvature, of a curve      98
Curvature, of a curve, $\delta_{b}^{a}$      8
Curvature, tensor      188
Derivative, covariant      189
Derived group      132Ч135 154 164 169 173 184
Dickson      15 91 294
Differential equations of a group      20
Differential equations of definition of a group      56 57
Differential equations of second order      100 107 108
Differential equations, admitting a group      82 86 88 89 95 96 100 101 107 135
Differential equations, admitting linear operators      85Ч91 106Ч108
Differential equations, associated      5 9
Differential equations, complete      7 8 12 86
Differential equations, completely integrable      1 4
Differential equations, linear partial      7 12 63 82 85Ч88
Differential equations, mixed      4
Differential equations, ordinary of first order      89 95Ч97
Differential equations, Pfaffian      93Ч95
Differential equations, system of      1 7
Differential equations, total      1
Differential invariant      96Ч98 100Ч107
Differentiation, covariant      189 195
Differentiation, covariant,      232 233
Dilatation      42
Direct product of groups      120Ч122 174 184
Displacement, parallel      199
Distance      191
Dummy index      7
Dynamics      277Ч280
Dynamics, $e$      186
Dynamics, $\eta_{j}^{i}(u)$      60 152
Eiesland      236 295
Einstein      196 295
Einstein space      206
Eisenhart      80 199 236 239 294Ч296
Elementary, divisor      146
Elementary, hyperplane      238
Elementary, hypersurface      267
Engel      54 91 164 293 296
Envelope      244 245 268 271 272
Equations, of a group      13 16 20 56 57
Equations, of a motion      209
Equipollent vectors      201Ч203 207 208
Equivalent function-groups      287 291
Equivalent groups      34 76Ч80 84 115
Equivalent points      67
Essential parameters      9Ч12 56
Euclidean space      186 188 190 191
Euclidean space,      208 212 218
Exceptional $G_{1}$ of a $G_{r}$      113 118 152 162 183
Extended group      92Ч97 107 252
Extension of a group      97Ч101 105 125
Factor group      129Ч131 137
FermatТs principle      280
Fine      13 15 243 246 258 264 294 295
Finsler      277 294
Flat space      188 193 203Ч205
Franklin      91 295
Frobenius      183 294
Fubini      217 218 221 229 294
Function group, basis of      282 285Ч291
Function group, commutative      283 287 288 290
Function group, equivalent      287 291
Function group, homogeneous      287Ч292
Function group, non-homogene- ous      281Ч287 291 292
Function group, reciprocal      283 285 288 290 291
Function group, subgroup of      282Ч286 290
Functions, commutative      283
Functions, in involution      250 283
Functions, singular      283 284 287 291 292
Fundamental theorem,      216 277
Fundamental theorem, $g^{ij}$      188 189 194 195
Fundamental theorem, $G_{2}$      60 61 90 91 135 137 155 184 228
Fundamental theorem, $G_{3}$      66 67 123 133 137 138 154 184 227 236 255
Fundamental theorem, $G_{4}$      66 123 136 137 155 184
Fundamental theorem, $g_{ij}$      156 174 186Ч189 194 195 205
Fundamental theorem, $G_{r}$      17
Fundamental theorem, $\Gamma_{\beta\gamma}^{\alpha}$      44 194 199 206 231Ч233
Fundamental Theorem, First      24
Fundamental Theorem, Second      54
Fundamental theorem, third      55
Fundamental,      208 226 236
Fundamental, quadratic form      186
Fundamental, tensor      187 195
Fundamental, variety      244
Fundamental, vector      148 166 170
Generic, rank      49 64
Generic, root      156 160
Geodesic      191 193 206 207 209Ч212 216 217 235 274Ч277 279
Geodesic, correspondence      236
Geodesic, hypersphere      275 279
Geodesic, sub-space      203
Geodesically parallel hypersunaces      211 235
Graham      182
Group      14
Group, admits a transformation      110 112 121
Group, basis of      28
Group, finite continuous      16 17
Group, generator of      40
Group, infinite      15 249
Group, infinitesimal transformation of a      36 40 41
Group, invariant under a transformation      110Ч112 121
Group, mixed      58
Group, one- parameter      32Ч36
Group, rank of a      159 160
Group, semi      15 17 18 21 28 29 31
Group, structure of a      28
Group, symbols of a      28
Group, trajectories of a      35 40 44 45 253 274 affine asystatic derived extended extension factor imprimitive induced integrable intransitive linear motion primitive projective semi-simple simple systatic transitive)
Group-space      27 31 44Ч47 198Ч206
Group-space,      213 229Ч231 236
Groups with the same structure      28 29 32 76 78 80 93 98 125 equivalent isomorphism reciprocal)
Groups, classification of      165 180Ч183
Groups, classification of, product of      120Ч122 174 184
Groups, similar      76
Hamiltonian equations      263 279
Hamiltonian equations, function      263 279
Holonomic system      263 277
Homothetic transformation      42
Huntington      15 294
HuygenТs principle      272
Hyperplane      238
Hyperplane, tangential      239 267
Hypersphere      275 279
Hypersurface      210 237
Hypersurface of origin      271 272
Hypersurface, elementary      267
Hypersurface, geodesically parallel      210 211 235
Identity      15
Identity of Bianchi      232
Identity of Jacobi      6 250
Identity of Ricci      189 232
Identity, transformation      14
Imprimitive group      80Ч83 117 123 137
Imprimitivity, system of      80Ч85 117 137
Independent (cpnstant coefficients)      21
Indices of composition      128
Induced group      70Ч72 84 85 119 123 124 137Ч139
Infinitesimal, motion      208 209 231
Infinitesimal, motion, transformations of a group      36 40 41
Integrable group      134Ч137 144 145 162Ч165 172 173 183 184
Intransitive group      71Ч73 78Ч80 98
Intransitive group,      210 216Ч226 235 236
Invariant of points      108
Invariant sub-group      118Ч124 129Ч134 137Ч139 144 153 154 158 161Ч163 173 174 183 184 203 235
Invariant sub-group, maximum      122 127Ч131
Invariant variety      67Ч72 80 81 85
Invariant variety,      123 138 153 210 211
Invariant variety, isolated      119
Invariant variety, minimum      67Ч71 211 221 235
Invariant vector-spaces      142Ч145 163
Invariant, absolute, of a group      61 62 67Ч69 93 95 96 98 107 159 161
Invariant, differential      96Ч98 100Ч107
Invariant, direction      140 142 144
Invariant, numerical      291
Invariant, relative, of a group      63 66 67 69 93 95 107
Inverse transformation      13 249
Inverse transformation of a semi-group      17
Involution, functions in      250 283
Isolated variety      119
Isomorphism of groups, simple      124Ч127 131 137 151 208
Isomorphism of groups, simple, multiple      124Ч127 129 136 151 184
jacobi      263 293
Jacobi, form of a complete system      8
Jacobi, identity      6 250
Jacobi, relations      26
Killing      158 164 180 183 184 208 293
Killing, equations of      208 215Ч221 237
Knebelman      236 295 296
Kronecker delta      8
Kronecker delta, $L_{\beta\gamma\delta}^{\alpha}$      198
Kronecker delta, $L_{\beta\gamma}^{\alpha}$      29 198 199 229
Kronecker delta, $\Lambda_{\beta\gamma\delta}^{\alpha}$      75 193 195
Kronecker delta, $\Lambda_{\beta\gamma}^{\alpha}$      74 192Ч197 217 218 230
Lagrangian function      263 278
Length, element of      186
Levi Ч Civita      190 267 294 295
Lie      36 39 54 55Ч58 66 80 84 91 106Ч108 119 122 123 131 137 145 154 155 163 164 180 183Ч185 242 248Ч252 255 258Ч261 278 281 282 285 287-292 293
Lie group      40
Linear, complex      185
Linear, complex, group      43 138 143Ч150 153 163 184 234 292
Matrix, characteristic      155
Matrix, equation      142Ч144 148Ч150
Matrix, trace of a      157
Mattoli      194 295
Maurer      23 293
Maximum invariant sub-group      122 127Ч131
Metric, generalized      277
Metric, Riemannian      187 206
Michal      43 295
Minimum invariant variety      67Ч71
Minimum invariant variety,      221 223 235
Mixed, group      58
Mixed, system of differential equations      4
Moore      15 294
Motion, in 2Чspace      42 80 98 226Ч228
Motion, in 3Чspace      43 80 228
Motion, infinitesimal      209
Motions, group of      57 208Ч221 230Ч237
Motions, in a linearly connected manifold      229Ч235 237
Motions, intransitive      210 216Ч226 235 236
Motions, multiply transitive      221
Motions, of maximum order      215 224 236
Motions, simply transitive      218 235 236
Multiply isomorphic groups      see Isomorphism
Normal, covariant      239 242 243
Null vector      174 187 212
Null vector, $\Omega_{jk}^{i}$      195Ч198 205Ч207 231Ч235 237
Operator, linear      6
Parallelism, $(+)-$      199Ч205
Parallelism, $(-)-$      199Ч205
Parallelism, (0)$-$      199 204 205
Parallelism, absolute      190 193 194
Parallelism, absolute,      217 234 235
Parallelism, relative      190 191 193 199
Parameter affine      199
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