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| Bliss G.A. — Calculus of Variations |
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| Предметный указатель |
Abbatt 181
Barnett 80 182
Barrow 1
Bernoulli, James 10 54 175 181
Bernoulli, John 10 42 54 181 182
BLISS 183
Bolza 9
Borda 78 182
Brachistochrone problem 9 41—84 182
Brachistochrone problem, auxiliary formulas 62
Brachistochrone problem, construction of a field 55—62
Brachistochrone problem, descriptive summary of results 42—43
Brachistochrone problem, extremals are cycloids 50—54
Brachistochrone problem, first necessary condition 47
Brachistochrone problem, from a curve to a point 78
Brachistochrone problem, from a point to a curve 70
Brachistochrone problem, initial velocity zero 68
Brachistochrone problem, integral to be minimized 45
Brachistochrone problem, sufficient conditions for a minimum 66
Brachistochrone problem, the invariant integral 65
Byerly 181
Catenaries 85 89
Catenaries, envelope of the family 86 97
Catenaries, family through a point 92 95
Catenaries, through two points 96
Cauchy 3
Cavalieri 1
Clebsch 179
Conjugate points for catenaries 86 94 103
Conjugate points, in general 131 148 165
Corner conditions for brachistochrones 50
Corner conditions, for catenaries 92
Corner conditions, in general 143
Cycloids 42 52
Cycloids, through two points 55
Darboux 87 178 182
Descartes 1
Dienger 181
Dirksen 181
Du Bois Reymond 183
Envelope theorems for brachistochrones 72
Envelope theorems for shortest distances 32
Envelope theorems, for catenaries 87 102 124
Envelope theorems, for more general cases 131 140 168
Envelope theorems, references 183
Erdmann’s comer condition 143 183
Euler 48 175—76 181 182
Euler, his equation 48 130
Extremals for shortest distances 22
Extremals, for minimum surfaces of revolution 89
Extremals, for more general cases 131 145
Extremals, for the brachistochrone problem 50
Fermat 1
Fields of extremals for shortest-distance problems 27 33
Fields of extremals, for minimum surfaces of revolution 104 112 124
Fields of extremals, for more general cases 100 132 151 154 172
Fields of extremals, for the brachistochrone problem 57 76
Fields of extremals, properties of field functions 60 106 152 156
Focal points 79 124 169
Focal points, geometric constructions 79 125
Fundamental auxiliary, formulas for shortest-distance problems 23
Fundamental auxiliary, formulas, for more general cases 98 136
Fundamental auxiliary, formulas, for the brachistochrone problem 62
Fundamental lemma 20 182
Galileo 13 42 174 182
General theory 128—179
General theory, analytic proof of Jacobi’s condition 161
General theory, construction of a field 154—157
General theory, descriptive summary of results 130—136
General theory, formulation of the problem 128
General theory, fundamental auxiliary formulas 98 136
General theory, fundamental lemma 20 182
General theory, historical re-remarks 174
General theory, necessary conditions for a minimum 47 130 136 138 140 143
General theory, regular problems 154 160—161
General theory, sufficient conditions for a minimum 133 157—159
General theory, the envelope theorem 131
General theory, the extremals 145
General theory, when both end-points are variable 172
General theory, when one end-point is variable 166
Goldschmidt 88
Goldschmidt, his discontinuous solution 111 116 182
Goursat 180 181
Goursat — Hedrick 183
hadamard 178 180 181
Hahn 180
| Hancock 181
Hilbert’s differentiability condition 144 178 183
Hilbert’s invariant integral for shortest-distance problems 25
Hilbert’s invariant integral, for minimum surfaces of revolution 108
Hilbert’s invariant integral, for more general problems 100 141 178 183
Hilbert’s invariant integral, for the brachistochrone problem 64
Historical remarks 174
Huygens 53
Isoperimetric problems 15
jacobi 41 87 177 181 182
Jacobi, his necessary condition for shortest-distance problems 32
Jacobi, his necessary condition, an analytic proof 161
Jacobi, his necessary condition, for minimum surfaces of revolution 87 102 124
Jacobi, his necessary condition, for more general cases 132 140 169
Jacobi, his necessary condition, for the brachistochrone problem 74
Jellett 181
Kneser 87 178 180 181 182
Lagrange 78 174 181 182
Lecat 179 180
legendre 131 176 181
Legendre, his necessary condition 138
Leibniz 2 3
Light paths 14
Lindeloef 181 183
Lindeloef, his conjugate point construction 94 102 183
l’Hospital 11
Maclaurin 5
MacNeish 89 118 182 183
Maxima and minima 3
Maxima and minima, criteria 4
Mayer 179
Moigno — Lindeloef 181
Moore, E.H. 182
Necessary conditions for a minimum for shortest-distance problems 18 33
Necessary conditions for a minimum, for brachistochrone problems 50—51
Necessary conditions for a minimum, for minimum surfaces of revolution 92 103 124
Necessary conditions for a minimum, for more general cases 48 130—131 136 143—145 161 166 173
Newton 2 3 174
Newton, his resistance problem 8 182
notes 182
Ohm 181
Osgood 183
Ostwald’s Klassiker 182
Pascal, B. 1
Pascal, E. 181
references 180
Relative minima 134
Relative minima, sufficient conditions 154
Riemann 3
Roberval 1
Schwarz 43
Shortest distances 17—40
Shortest distances, between two curves 38
Shortest distances, between two points 17—30
Shortest distances, extremals are straight lines 21
Shortest distances, from a point to a curve 30
Shortest distances, from a point to an ellipse 34
Shortest distances, sufficiency proofs 21 27 33—34
Sinclair 80 89 121 126 182 183
Soap films 7 119 126
Stegmann 181
Strauch 181
Sufficiency theorems for shortest-distance problems 21 27 33
Sufficiency theorems, for brachistochrone problems 66 75
Sufficiency theorems, for minimum surfaces of revolution 108 112 124
Sufficiency theorems, for more general cases 133 151 154 169
Surfaces of revolution of minimum area 7 85—127
Surfaces of revolution of minimum area, continuity of the extremal integral 113
Surfaces of revolution of minimum area, descriptive summary of results 85—89
Surfaces of revolution of minimum area, generated by straight lines 110
Surfaces of revolution of minimum area, necessary conditions for a minimum 89 102 124
Surfaces of revolution of minimum area, sufficient conditions for a minimum 109 124
Surfaces of revolution of minimum area, the absolute minimum 115
Surfaces of revolution of minimum area, with one end-point variable 122—126
Todhunter 180 181
Tonelli 179 181
Transversality conditions for shortest distances 32
Transversality conditions, for minimum surfaces of revolution 123
Transversality conditions, for more general cases 167
Transversality conditions, for the brachistochrone problem 72
Wallis 1
Weierstrass 3 132 177 183
Weierstrass, his comer condition 143 183
Weierstrass, his necessary condition 131 138
Weierstrass, his sufficiency proofs 27 66 88 108 124
Woodhouse 181
Zermelo 87 178 189 182
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