Bolza O. — Lectures of the Calculus of Variations :: Электронная библиотека попечительского совета мехмата МГУ

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Bolza O. — Lectures of the Calculus of Variations

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Название: Lectures of the Calculus of Variations

Автор: Bolza O.

Аннотация:

My principal source of information concerning Weier-strass's theory has been the course of lectures on the Calculus of Variations of the Summer Semester, 1879, which I had the good fortune to attend as a student in the University of Berlin. Besides, I have had at my disposal sets of notes of the courses of 1877 (by Mr. Gr. Schulz) and of 1882 (a copy of the set of notes in the "Lesezimmer" at Gottingen for which I am indebted to Professor Tanner), a copy of a few pages of the course of 1872 (from notes taken by Mr. Ott), and finally a set of notes (for which I am indebted to Dr. J. C. Fields) of a course of lectures on the Calculus of Variations by Professor H. A. Schwarz (1898-99). - Oskar Bolza

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1

Год издания: 1904

Количество страниц: 271

Добавлена в каталог: 18.04.2010

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Предметный указатель

Absolute maximum, minimum      10
Accumulation-point, of a set of points      178 254
Admissible curves      9 11 104 121 206
Amplitude, of a vector      9
Bliss's condition, for the case of two variable end-points      113
Boundary conditions, along segment of boundary      43 149
Boundary conditions, at points of transition      42 150 267
Boundary conditions, when minimizing curve has one point in common with boundary      152 267
Boundary, of set of points      5
Brachistochrone      126 135 146
Brachistochrone, case of one variable end-point      106
Brachistochrone, determination of constants      128
Catenoid      see "Surface of revolution of minimum area"
Circle, notation for      9
Class C, C', C''....D', D''.., curves of      8 116
Class C, C', C''....D', D''.., curves of class K      161
Class C, C', C''....D', D''.., functions of      7
Closed region      5
Closed set of points      178 267
Co-ordinates: agreement concerning positive direction of axes      8
Conjugate points      60
Conjugate, case where the two end-points are conjugate      65 204
Conjugate, for isoperimetric problems      221
Conjugate, for the case of parameter-representation      135
Conjugate, geometrical interpretation      63 137
Connected set of points      5
Continuous functions, continuity of compound functions      21
Continuous functions, definitions and theorems on: existence of maximum and minimum      13 80
Continuous functions, integrability      12
Continuous functions, sign      21
Continuous functions, uniform continuity      80
Continuum      5
Convex region      247
Corner, corner-conditions      38 126 210
Corner, defined      8 117
Critical point      109
Curves of class C', C''      116
Curves of class C, C',.. D'      8
Curves of class K      161
Curves, (a) representable in form y=f(x)      8
Curves, (b) in parameter-representation      115
Curves, Jordan curves      180
Curves, ordinary      117
Curves, rectifiable      116
Curves, regular      117
Curvilinear co-ordinates, in general      181
Curvilinear co-ordinates, Kneser's      184
Definite integrals, connection with indefinite integral      89
Definite integrals, differentiation with respect to a parameter      16
Definite integrals, first mean-value theorem      24
Definite integrals, integration by parts      20
Definite integrals, theorems on: integrable functions      12 89
Derivatives, notation      6 7
Derivatives, progressive and regressive      7
Derivatives, reversion of the order of differentiation in partial derivatives of higher order      18
Differential equations, dependence of the general integral upon the constants of integration      54
Differential equations, existence theorem      28
Differential equations, upon parameters      71 223
Discontinuous solutions      36 125 209
Distance: between two points, notation      9
Domain      5
End-points, variable      see "Variable end-points"
Envelope of a set of extremals      62
Envelope of a set of plane curves in general      62 137
Envelope, case when the envelope degenerates into a point      204
Envelope, case when the envelope has cusps      201
Envelope, extension of this theorem to extremals      174
Envelope, theorem on the envelope of a set of geodesics      166
Equilibrium, of cord suspended at its two extremities      211 231 241
Equivalent problems      183 197 228
Erdmann's corner condition      38
Euler's (differential) equation      22;
Euler's (differential) equation, assumptions concerning its general integral      54 130;
Euler's (differential) equation, cases of reduction of order      26 29
Euler's (differential) equation, Du Bois-Reymond's proof of      23
Euler's (differential) equation, Hilbert's proof of      24
Euler's (differential) equation, Weierstrass's form of      123
Euler's isoperimetric rule      210
Evolute, of plane curve      174
Existence theorem for a minimum "im Grossen,"      245
Existence theorem for a minimum "im Kleinen,"      146
Existence theorem for differential equations      28
Existence theorem, in particular for linear differential equations      50
Extraordinary vanishing of the E-function      142
Extremal, construction of extremal through given point in given direction      28 124
Extremal, construction of extremal through two points, sufficiently near to each other      146
Extremal, defined      27 123 209
Extremal, problems with given extremals      30
Extremal, set of extremals cut transversely by a given curve      111
Extremal, set of extremals through given point      60
Extremum, defined      10 compare "Maximum"
Field for case of parameter-representation      144 176
Field for isoperimetric problems      241
Field, applied to set of extremals through $\overline{A}$       82
Field, defined      79
Field, field-integral      266
Field, improper      83
Field, theorem concerning existence of      79
First necessary condition      see "Euler's differential equation"
First variation for case of parameter-representation      122 123
First variation for case of variable end-points      102
First variation for isoperimetric problems      209
First variation, defined      17
First variation, transformation by integration by parts      20 22
First variation, vanishing of the      18
Focal point: of a transverse curve on an extremal, according to Kneser      200
Focal point: of a transverse curve on an extremal, case where end-point B coincides with focal point      204
Focal point: of a transverse curve on an extremal, defined      109
Focal point: of a transverse curve on an extremal, equation for its determination, according to Bliss      108 155
Focal point: of a transverse curve on an extremal, geometrical interpretation      111 156
Fourth necessary condition      see under "Weierstrass"
Free variation, points of      41
Function $F_{1}$       121
Function $F_{2}$       132
Function $\textbf{E}(x, y; p, q; \widetilde{p}, \widetilde{q})$ , defined      138
Function $\textbf{E}(x, y; p, q; \widetilde{p}, \widetilde{q})$ , homogeneity properties      140
Function $\textbf{E}(x, y; p, q; \widetilde{p}, \widetilde{q})$ , Kneser's geometrical interpretation      195
Function $\textbf{E}(x, y; p, q; \widetilde{p}, \widetilde{q})$ , ordinary and extraordinary vanishing      142
Function $\textbf{E}(x, y; p, q; \widetilde{p}, \widetilde{q})$ , relation between E-function and $F_{1}$       141
Function $\textbf{E}(x, y; p, \widetilde{p})$ , defined      34 75
Function $\textbf{E}(x, y; p, \widetilde{p})$ , geometrical interpretation of this relation      77
Function $\textbf{E}(x, y; p, \widetilde{p})$ , relation between $\textbf{E}(x, y; p, \widetilde{p})$ and $F_{y'y'}$       76
Function $\textbf{E}_{1}(x, y; p, q; \widetilde{p}, \widetilde{q})$       145
Function $\textbf{E}_{1}(x, y; p, \widetilde{p})$       76
Fundamental lemma, of the Calculus of Variations      20
Generalized integral      157 248 compare
Geodesic curvature      129
Geodesic distance      176
Geodesic parallel co-ordinates      164
Geodesics      128 146 155
Geodesics, Gauss's theorems on      164 165
Geodesics, theorem on the envelope of a set of      166
Hilbert's construction      253
Hilbert's existence theorem      245
Hilbert's invariant integral      92 195
Homogeneity condition      119
Homogeneity, consequences of      120
Implicit functions, theorem on      35
Improper field      83
Improper maximum, minimum      11
In a domain, use of the word explained      5 6
Infinitesimal      6
Inner point      5
Integrability condition      29
Integrable functions, theorems on      12 89
Integral for case of parameter-representation      117
Integral, condition for invariance under parameter-representation      119
Integral, extension to curves without a tangent, (a) Weierstrass's      157
Integral, extension to curves without a tangent, (b) Hilbert — Osgood's      248
Integral, taken along a curve, definition and notation      8

Integration, by parts      20 20
Interval, defined      5
Invariance, of E and $F_{1}$       183
Isoperimetric constant      209
Isoperimetric constant, Mayer's theorem for case of discontinuous solutions      209
Isoperimetric problems, in general      206—244
Isoperimetric problems, special      4 210 229 238
Isoperimetric problems, with variable end-points      113
Jacobi's condition      67
Jacobi's condition for case of one variable end-point      109 155 200
Jacobi's condition for isoperimetric problems      225 226
Jacobi's condition, Kneser's form of      136
Jacobi's condition, proofs of its necessity      65 66
Jacobi's condition, Weierstrass's form of      135
Jacobi's criterion      60 135
Jacobi's differential equation      49 133
Jacobi's theorem concerning the integration of Jacobi's differential equation      54 135
Jacobi's transformation of the second variation      51
Jacobian      57
Jordan curve      180
Kneser's curvilinear co-ordinates      184
Kneser's sufficient conditions      187
Kneser's theorem on transversals      172
Kneser's theory      164—205
Lagrange's differential equation      22
Legendre's condition      47
Legendre's condition, for isoperimetric problems      217
Legendre's condition, Legendre's differential equation      46
Legendre's condition, Weierstrass's form of      133
Length of a curve, Jordan's definition      157
Length of a curve, Peano's definition      249
Limit attained by continuous function      13 80
Limit, criterion for the existence of      258
Limit, definition and notation      7
Limit, lower and upper      3 10
Limit, uniform convergence to a      19
Limit-point      see "Accumulation-point"
Limited variation, functions of      258
Lindeloef's construction      64
Linear differential equations of the second order, Abel's theorem      58
Linear differential equations of the second order, existence theorem      50
Linear differential equations of the second order, Sturm's theorem      58
Lower limit      3 10
Maximum      see "Minimum"
Mayer's law of reciprocity for isoperimetric problems      229 244
Mean-value theorem, first, for definite integrals      24
Minimum for case of parameter-representation      121
Minimum Hilbert's a-priori existence proof of a minimum "im Grossen,"      245—263
Minimum of a continuous function      13 80
Minimum of a definite integral, absolute and relative      10
Minimum, existence of a minimum "im Kleinen,"      146
Minimum, proper and improper      11
Minimum, semi-strong in case of isoperimetric problems      244
Minimum, weak and strong      69 70
Neighborhood of a curve      10
Neighborhood $(\rho)$ of a curve      13 121
Neighboring curve      14
Numerable set of points      261 268
One-sided variations      see also "Boundary conditions"
One-sided variations, analytic expression for      42 148
One-sided variations, necessary conditions for a minimum with respect to      42 149
One-sided variations, sufficient conditions      42
Open region      5
Ordinary curves, defined      117
Ordinary vanishing of the E-function      142 266
Osgood's theorem concerning a characteristic property of a strong minimum      190
Parameter representation, curves in      115
Parameter-transformation      116
Partial derivatives      see "Derivatives"
Partial variation, of a curve      37
Point of a set      12
Point-by-point variation, of a curve      41
Positively homogeneous      119
Progressive derivative      7
Proper minimum      11
Rectifiable curves      116 250 251 251 251 compare
Region, closed      5
Region, defined      5
Region, open      5
Regressive derivative      7
Regular curves      117
Regular functions      21
Regular problems      29 40 97 125
Relative maximum or minimum      10 10
Second necessary condition      see "Legendre's condition"
Second variation      44—67
Second variation for case of variable end-points      102
Second variation for case of variable end-points in parameter-representation      102 155
Second variation for isoperimetric problems      216—225
Second variation, Jacobi's transformation of      51
Second variation, Legendre's transformation of      46
Second variation, Weierstrass's transformation of, for case of parameter-representation      131
Semi-strong extremum      244
Semi-strong extremum, sufficient conditions for      244
Set of points, accumulation points of      178
Set of points, boundary point of      5
Set of points, closed      178 267
Set of points, connected      5
Set of points, continuum      5
Set of points, definition      10
Set of points, inner point of      5
Set of points, numerable      261 268
Set of points, upper and lower limits of one-dimensional set      3 10
Sign of square roots, agreement concerning      2
Slope restrictions      101
Solid of revolution, of minimum resistance      73 142
Strong extremum, defined      70
Strong extremum, sufficient conditions for      see "Sufficient conditions"
Strong variation      72
Sturm's theorem, on homogeneous linear differential equations of the second order      58
Substitution symbol      5 6
Sufficiency proof, for geodesics      165
Sufficient conditions for strong minimum for case of parameter-representation, Weierstrass's      143—146
Sufficient conditions for strong minimum for isoperimetric problems, Weierstrass's      237 243
Sufficient conditions for strong minimum for one-sided variations      42
Sufficient conditions for strong minimum, extension to curves without a tangent, Weierstrass's proof      161
Sufficient conditions for strong minimum, in case of one movable end-point      109
Sufficient conditions for strong minimum, in case of two movable end-points      113
Sufficient conditions for strong minimum, in terms of $F_{y\ y'}$       96
Sufficient conditions for strong minimum, Kneser's sufficient conditions for case of one movable end-point      187
Sufficient conditions for strong minimum, Osgood's proof      192
Sufficient conditions for strong minimum, when x independent variable, in terms of E-function      95
Sufficient conditions for weak minimum      70
Surface of revolution of minimum area      1 27 48 64 97 153
Taylor's theorem      14
Third necessary condition      see "Jacobi's condition"
Third variation      59
Total differential      25
Total variation      14
Transversal, degenerate      169
Transversal, Kneser's theorem on transversals      172
Transversal, to set of extremals      168
Transverse for isoperimetric problems      210
Transverse in parameter-representation      155
Transverse, condition of transversality      36 106
Transverse, curve transverse to an extremal      106
Unfree variation, points of      41
Uniform continuity      80
Uniform convergence, to a limit      19
Upper limit      3 10
Variable end-points, case when both end-points movable on given curves      113
Variable end-points, general expression of first variation for case of      102
Variable end-points, of second variation      102
Variable end-points, one end-point fixed, the other movable on given curve, treated (a) by the method of differential calculus      102—113
Variable end-points, one end-point fixed, the other movable on given curve, treated (b) by Kneser's method      164—205 for "Focal "Sufficient
Variation for case of parameter-representation      122 122
Variation of a curve      14
Variation of type $\omega(x, \epsilon)$       18
Variation special variation of type $\epsilon\eta$       15
Variation, definition for first, second, etc.      16

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