Calculus of Variations and Partial Differential Equations of First Order

Front Cover
Preface To The Second English Edition
Preface To The First English Edition
Preface To The First English Edition
Preface To The First German Edition
Biographical Note
List Of Examples
Table Of Contents
Part I Partial Differential Equations Of The First Order
1 Continuous Convergence, Implicit Functions, Ordinary Differential Equations
1. Continuous Convergence
2.
3. Normal Families.
4.
5. The Weierstrass Theorem.
6.
7. Necessary Conditions For The Extremum.
8. The Convolution Process.
9. Implicit Functions.
10.
11.
12.
13. Ordinary Differential Equations.
14.
15.
16.
17.
18.
19.
20. A Device Of Cauchy
2 Fields Of Curves And Multidimensional Surfaces, Complete Systems
21. Fields Of Curves.
22. Linear Homogeneous Partial Differential Equations Of The First Order.
23. Fields Of Multidimensional Surfaces.
24.
25.
26.
27,
28.
29. The Integrals Of A Completely Integrable System Of Total Differential Equations.
30. The Bracket Symbol.
32. Complete And Jacobi Systems.
33.
34.
3 Partial Differential Equations Of The First Order, Theory Of Characteristics
35. Structure And Definition Of Characteristics.
36.
37.
38. Cauchy'S Relations.
39.
40. Existence Of Solutions.
41.
42. Uniqueness Of Solutions.
43. Integration Of The Partial Differential Equations H(Xi, S,.) = 0.
44. The Lagrange Brackets
45.
46.
47. Integration Of The Partial Differential Equation S1 + H(T, X;, S,;) = O.
48.
49. A Device Of Jacobi.
50. Complete Solutions Of A Partial Differential Equation Of The Firstorder
4 Poisson Brackets, Systems Of Partial Differential Equations Of The First Order
51. Poisson Brackets.
52.
53.
54..
55. Involutory Systems Of Functions.
56.
57. The Square Brackets.
58. Simultaneous Partial Differential Equations Of The First Order.
59. Involutory Systems.
60. Derivation Of The Equations For The Characteristics.
61. The Complete Integrability Of The Total Differential Equations For The Characteristics
62.
63. Theory Of Characteristics And Integration Of Involutory Systems.
64.
65. Involutory Systems Of Partial Differential Equations Without S.
66.
67.
5 Elements Of Tensor Calculus
68. Definition And Purpose Of Tensor Calculus.
69. Scalar Quantities.
70. Contravariant Vectors.
71.
72. Covariant Vectors.
73. The Inner Product Of Vectors.
74. Definition Of Tensors.
75.
76. Contraction.
77. Tensors Of Higher Rank.
78. Tensor Calculus. Formation Of General Products
79.
80. Tensor Analysis.
81.
82. Application To Systems Of Linear Homogeneous Partial Differential Equations.
83. Application To The Equations Of Mechanics.
84.
85.
6 Canonical Transformations
86. Definition And Examples.
87.
88.
89. Definition Of Canonical Transformations By Lagrange Brackets.
90.
91. Definition Of Canonical Transformations By Poisson Brackets.
92.
93.
94. The Group Property.
95. Elementary Canonical Transformations.
96. A Property Of The Matrix ((?Xi/?Xj), (?I/?Yj))
97. Representation With The Help Of A Generating Function.
98.
99.
100.
101.
102.
103.
104.
105. The Most General Canonical Transformation With Prescribed X;(Xi, Yi).
106.
107. Normalization Of The Generating
109. Determination Of Canonical Transformations With A Prescribed Function ?.
110. Homogeneous Canonical Transformations.
111.
112.
113. Families Of Canonical Transformations.
114.
7 Contact Transformations
116. The Concept Of A Union Of Elements.
117.
118.
119. Definition Of Contact Transformations.
120. The Relation Of Canonical To Contact Transformations.
121.
122.
123. Poisson And Square Brackets.
124.
125.
126.
127.
128.
129. The Invariance Property Of Contact Transformations.
130. Contact Transformations In (X, P).
131. Representation Of Contact Transformations By A Generating Function.
132.
133.
8 The Pfaff Problem
134. Statement Of The Problem.
135. Reduction Of Pfaffian Forms.
136.
137.
138. Equivalent Pfaffian Forms.
139. Formation Of Bilinear Covariants.
140. Singular Line Elements. The Class Of A Pfaffian Form.
141.
142.
143. Families Of Singular Line Elements.
144. Application To The Theory Of Partial Differential
145. Coupled Pfaffian Forms.
146:
9 Function Groups
147. The Generalized Bracket Symbols.
148.
149.
150. Change Of Variables.
151. Subgroups. Polar Groups.
152.
153. Union And Intersection Of Subgroups.
154.
155. Involutory Subgroups.
156. Normalization Of The Group.
157.
159. Distinguished Functions Of Groups.
160.
161.
162.
10 The Integration Theories Of Lagrange, Jacobi, Adolph Mayer And Lie
163. Introduction.
164. Complete Integrals.
165. The Theorem Of Jacobi
166.
167.
168. The Integration Theory Of Lagrange.
169.
170. The Jacobi Method Of Integration.
171.
172.
173. Invariance In Canonical Transformations.
174.
175.
176.
177.
178.
179. The Lie Method Of Integration.
180. The Poisson Theorem. The Last Method Of Integration Of Lie.
181. The Theorem Of Adolph Mayer.
182.
183.
184.
Supplement
Part Ii Calculus Of Variations
11 Ordinary Maxima And Minima. Quadratic Forms
185. Statement Of The Problem. Extremum In The Small.
186. Necessary Conditions For The Extremum On A Hypersurface.
187.
188. Quadratic Forms.
189.
190.
191.
192.
193.
194. Orthogonal Transformations.
195.
196.
197. Bounds For The Roots Of ?M(?)
198.
199. Positive Definite And Semi-Definite Quadratic Forms.
200.
201.
202.
203.
204.
205.
206.
207. Classification Of Quadratic Forms.
208. Sufficient Conditions For The Minimum In The Small.
209.
210.
211.
212.
213.
214. Reciprocal Quadratic Forms.
215.
216,
217.
12 Simple Variational Problems In The Small
218. The Concept Of The Close Neighborhood.
219. Definition Of Extremals.
220. Smoothing Of The Corners.
221.
222. Estimation Of A Special Variation.
223.
224.
225. Classification Of Line Elements.
226. The Function L1(T, X,, X,).
227. Equivalent Variational Problems.
228. Proof Of The Existence Of Extremals.
229.
230.
231,
232. The Fundamental Equations Of The Calculus Of Variations.
233. The Weierstrass ?-Function.
234.
235. The Hamiltonian Function.
236. The Legendre Condition And The Weierstrass ?-Function In Canonicalcoordinate
237.
238. Variational Problems In Canonical Coordinates.
239. Integration Of The Fundamental Equations.
240. The Euler Differential Equations.
241.
242. Definition And Properties Of Fields Of Extremals.
243.
244. Determination Of All Regular Extremals.
245.
246. The Intersection Points Of Neighboring
13 Variational Problems In Parametric Representation
247. Weierstrass'
Formulation.
248.
249. Homogeneity
250.
251. The Legendre Condition
252. The Function F1
253. Hamiltonian Functions.
254.
255.
256. The Legendre Condition In Canonical Coordinates.
257.
258. Calculation Of The Basic Function F From A Hamiltonian Function.
259.
260.
261. The Fundamental Equations In Parametric Representation.
262.
263.
264. Extremals And Fields Of Extremals.
265.
266.
267. The Fundamental Formula And Strong Extremals.
268.
269.
270. Concept Of The Extended Neighborhood.
14 Positive-Definite Variational Problems
271. Examples Of Plane Variational Problems.
272. The Brachistochrone.
273. Minimum Surface Of Revolution.
274. Example Of A Line Integral Whose Value Depends On The Sense Of Traversal
276. Quickest Nautical Path For A Given Stationary Sea Current.
277.
278.
279.
280.
281.
282.
283.
284.
285.
286.
287.
288. General Properties Of Positive-Definite Variational Problems.
289. The Lndicatrix.
290.
291.
292. Regular Variational Problems.
293.
294. The Figuratrix.
295.
296.
297. Transversality.
298. Geodesically Equidistant Surfaces.
299.
300.
15 Quadratic Variational Problems
Theory Of The Second Variation
301. Inhomogeneous Variational Problems Independent Of T
302.
303.
304. Quadratic Variational Problems.
305.
306.
307. The Maupertuis Principle.
308.
309. Riemann Spaces. Christoffel Symbols.
310.
311. Remark On The Legendre Condition.
312.
313. The Jacobi Differential Equations.
314.
315. The Transformation Of The Second Variation.
317. Field-Like Structure Of The Original And The Accessory Variational Problem.
318.
319.
320. Normalization Of The Basis Of A Field-Like Structure Of The Accessory Problem.
321.
322.
323.
32,T. Properties Of The Field-Like Families Of Extremals Of The Accessory Problem.
325.
326.
327. Focal Points And Focal Surfaces.
328.
329. Construction Of A Field In The Large.
330. Distinguished Fields Of Extremals
331. The Tonelli Construction.
332. Theory Of Conjugate Points.
333.
334.
335.
336. The Sturrn-Liouville 011cillation Theorems.
337.
338.
339.
340. Example. Surface Of Revolution Of Smallest Area
341.
342. Generalized Geodesic Lines
343.
344.
345.
346.
347.
348.
349. Generalization Of A Theorem Of Euler.
350.
351.
352.
353. The General Theorem On Envelopes.
354.
355. The Jacobi Theorem On Envelopes.
356.
16 The Boundary-Value Problem And The Question Of The Absolute Minimum
357. The Classical Representation Of The Problem In The Calculus Of Variations.
358. The Boundary-Value Problem For The Euclidean Plane And The Bernoulli Brachistochrone.
359.
360. The Boundary- Value Problem For The Surface Of Revolution Of Smallest Area.
361.
362.
363.
364.
365.
366.
367.
368. Example Of A Regular Variational Problem For Which The Problem Of The Absolute Minimum Is Not Always Solvable.
369.
370.
371. An Example Of Euler.
372. Singular Line Elements Which Have The Character Of Irregular Line Elements.
373.
374. Further Examples. Singular Extremals.
375. An Example Of Maxwell.
376. Singular Strong Extremals.
377.
378. The Boundary-Value Problem In The Small.
379.
380.
381.
382.
383. The Boundary-Value Problem In The Small For Regular Variational Problems.
384.
385. The Boundary- Value Problem In The Large.
386.
387.
388.
389.
390.
391.
392.
393.
394. Examples.
395.
396.
397.
398.
399.
400.
401.
402. Regular Variational Problems On The Sphere.
403. Concluding Observations.
17 Closed Extremals. Periodic Variational Problems
404. Statement Of The Problem.
405.
406.
407. Normalization Of The Basic Function.
408.
409. A Sufficient Condition For The Occurrence Of A Minimum.
410.
411. Observation On Conjugate Points.
412. The Theorem Of Poincare.
413.
414. The Theory Of Hadamard And Razmadzli.
415.
416.
417.
18 The Problem Of Lagrange
418. Statement Of The Problem.
419
420.
421. Definition And Construction Of The Fields Of Extremals Of A Lagrange Problem
422.
423. Introduction Of Canonical Coordinates. Properties Of The Hamiltonian Function.
424.
425.
426.
427.
428.
429.
430. The Legendre Condition In Canonical Coordinates.
431. Extremal& And Fields Of Extremal&,
432. Weierstrass Parametric Representation.
433. A New Statement Of The Problem.
434.
435. Concept Of The Class Of An Extremal Arc.
436.
437.
438. The Clah Of Analytic Variational Problems.
439.
440.
441.
442. Determination Of The Dimension Of Distinguished Fields Of Extremals.
443.
444. Theory Of The Second Variation For Lagrange Problems Of Class Zero.
445. Problems With Class Q > 0.
446. Problems With Higher Derivatives.
447.
448.
449. The Problem Of Generalized Geodesic Lines.
450.
451. The Problems Of Mayer.
452.
453. Isoperimetric Problems.
454.
455.
456. A Special Isoperimetric Problem
457.
458. The Zermelo Navigation Problem.
459.
460.
461. The Delaunay Problem.
462.
463.
464.
465.
466.
467.
Guide To The Literature
Part I
Part Ii
Index
Back Cover