Lectures on the calculus of variations:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | Undetermined |
| Veröffentlicht: |
Chicago u. a.
Univ. of Chicago Pr.
1968
|
| Ausgabe: | 8. impr. |
| Schriftenreihe: | Phoenix science series
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Kopie, erschienen im Verl. Univ. Microfilms Internat., Ann Arbor, Mich. |
| Beschreibung: | IX, 292 S. |
Internformat
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| 245 | 1 | 0 | |a Lectures on the calculus of variations |
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Datensatz im Suchindex
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| adam_text | TABLE OF CONTENTS
PART I. SIMPLER PROBLEMS OF THE
CALCULUS OF VARIATIONS
CHAPTEI PAGE
I. The Calculus of Variations in Three Space 3
1. The nature of problems of the calculus of variations 3
2. The origin of the name calculus of variations 6
3. Analytic formulation of the problem 7
4. The first and second variations 9
5. The fundamental lemma 10
6. Necessary conditions from the first variation 11
7. Families of extremals 15
8. Auxiliary theorems 18
9. The necessary conditions of Weierstrass and Legendre .... 20
10. Envelope theorems and Jacobi s necessary condition 24
11. A second proof of Jacobi s condition 27
12. The determination of conjugate points 29
13. The geometric interpretation of conjugate points 34
II. Sufficient Conditions for a Minimum 37
14. Introduction 37
15. Auxiliary theorems 38
16. The sufficiency theorems of Weierstrass 39
17. A comparison of necessary with sufficient conditions 43
18. The definition and first properties of a field 43
19. A fundamental sufficiency theorem 45
20. Methods of constructing fields 46
21. Sufficient conditions for an integral to be independent of the path . 49
22. Further properties of the slope functions and extremals of a field . 51
23. The theory of the second variation 54
24. Sufficiency proofs without the use of fields 59
III. Fields and the Hamilton Jacobi Theory 65
25. Introduction 65
26. Canonical variables and canonical equations for extremals ... 65
27. A second proof of the imbedding theorem 68
28. Transversal surfaces of a field and the Hamilton Jacobi equation . 70
29. Extremals as characteristics of a partial differential equation . . 73
30. An application in dynamics 76
31. Extremals as curves of quickest descent 77
IV. Problems in the Plane and in Higher Spaces 81
32. Introduction 81
33. The problem in the plane 82
vii
viii TABLE OF CONTENTS
CHAPTEI PAGE
34. A comparison of the problems in the plane and in three space . . 83
35. The problem in a space of higher dimensions 86
36. The determination of conjugate points 88
37. The construction of fields 89
38. The Hamilton Jacobi theory 92
39. The theory of the second variation 97
V. Problems in Parametric Form 102
40. Introduction 102
41. Parametric representations of arcs 103
42. Formulation of the parametric problem 104
43. Consequences of the homogeneity relation 105
44. First necessary conditions for a minimum 108
45. The extremals Ill
46. The envelope theorem and Jacobi s condition 116
47. Analytic proof of the condition of Jacobi 118
48. The determination of conjugate points 119
49. Fields and a fundamental sufficiency theorem 124
50. Sufficient conditions for relative minima 127
51. Further sufficient conditions for strong relative minima .... 130
52. Canonical variables and equations 132
53. The imbedding theorem and the Hamilton Jacobi theory . . . 136
54. The construction of a complete integral of the Hamilton Jacobi
equation 141
55. Other theories of parametric problems 143
VI. Problems with Variable End Points 147
56. Introduction 147
57. Problems in three space with one end point variable on a surface . 148
58. Problems in three space with one end point variable on a curve . . 152
59. A more general problem with variable end points 157
60. A sufficiency theorem for the more general problem with variable end
points 158
61. The transversality condition 162
62. The second variation and a fourth necessary condition .... 163
63. Further sufficiency theorems 166
64. Another form of the fourth condition 167
65. Focal points for problems with one end point variable . . 170
66. Dependence of the focal point of a curve on curvature . . 175
67. Problems with variable end points in the plane 180
PART II. THE PROBLEM OF BOLZA
VII. The Multiplier Rule 187
68. Introduction 187
69. The equivalence of various problems 189
70. Analytic formulation of the problem of Bolza 193
71. Variations and the equations of variation 194
TABLE OF CONTENTS ix
CHAPTEI PACE
72. A fundamental imbedding lemma 196
73. The first variation of / 199
74. The multiplier rule 200
75. The extremals 206
76. Abnormality for minima of functions of a finite number of variables 210
77. Normality for the problem of Bolza 213
VIII. Further Necessary Conditions for a Minimum 220
78. The necessary conditions of Weierstrass and Clebsch 220
79. A lemma and a corollary 224
80. The second variation and a fourth necessary condition for a minimum 226
81. The accessory minimum problem 228
IX. Sufficient Conditions for a Minimum 235
82. Statement of the sufficiency theorem 235
83. An auxiliary theorem 236
84. Fields and their construction 237
85. A fundamental sufficiency theorem 240
86. The second variation for problems with separated end conditions
satisfying also the non tangency condition 243
87. The sufficiency theorem for problems with separated end conditions
satisfying also the non tangency condition 247
88. Sufficiency theorems for problems with end conditions unrestricted . 249
89. The second variation for problems with fixed end points .... 253
90. Conditions equivalent to the strengthened fourth condition . . . 257
91. Boundary value problems associated with the second variation . . 260
92. A sufficiency theorem applicable to some important abnormal cases . 264
APPENDIX
Appendix. Existence Theorems for Implicit Functions and Differential
Equations 269
I. Existence theorems for implicit functions 269
1. The fundamental existence theorem for implicit functions .... 269
2. An extension of the theorem of the preceding section 272
II. Existence theorems for differential equations 274
3. The existence of a solution through an initial point 274
4. Existence theorem for linear equations 276
5. The imbedding theorem 276
6. Derivatives with respect to constants of integration 278
BIBLIOGRAPHY
A Bibliography for the Problem of Bolza 285
INDEX
Index 291
|
| any_adam_object | 1 |
| author | Bliss, Gilbert Ames 1876-1951 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 8. impr. |
| format | Book |
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| indexdate | 2024-07-09T16:39:48Z |
| institution | BVB |
| language | Undetermined |
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| physical | IX, 292 S. |
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| spelling | Bliss, Gilbert Ames 1876-1951 Verfasser (DE-588)116202084 aut Lectures on the calculus of variations 8. impr. Chicago u. a. Univ. of Chicago Pr. 1968 IX, 292 S. txt rdacontent n rdamedia nc rdacarrier Phoenix science series Kopie, erschienen im Verl. Univ. Microfilms Internat., Ann Arbor, Mich. Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003833235&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bliss, Gilbert Ames 1876-1951 Lectures on the calculus of variations Variationsrechnung (DE-588)4062355-5 gnd |
| subject_GND | (DE-588)4062355-5 |
| title | Lectures on the calculus of variations |
| title_auth | Lectures on the calculus of variations |
| title_exact_search | Lectures on the calculus of variations |
| title_full | Lectures on the calculus of variations |
| title_fullStr | Lectures on the calculus of variations |
| title_full_unstemmed | Lectures on the calculus of variations |
| title_short | Lectures on the calculus of variations |
| title_sort | lectures on the calculus of variations |
| topic | Variationsrechnung (DE-588)4062355-5 gnd |
| topic_facet | Variationsrechnung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003833235&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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