Variational Analysis: Critical Extremals and Sturmian Extensions Book

Critical Extremals
Minimizing Extremals g: Fixed Endpoints     9
The Euler Equations     9
The Existence of Extremals Near g     13
The Necessary Conditions of Weierstrass and Legendre     17
The Jacobi Condition     19
Conjugate Points in the Nonsingular Case     23
The Hilbert Integral     31
Sufficient Conditions     39
The Algebraic Index of a Critical Extremal g     48
General End Conditions C[subscript r], Locally Defined     49
C[subscript r]-Transversality Conditions on g     53
The Second Variation of (J,C[subscript r]), r > 0     56
Critical Secondary Extremals: Endpoints Variable     64
Critical Secondary Extremals: Endpoints Fixed     73
Quadratic Index Forms, Derived and Free
Derived Index Forms     79
The Framed Family [Gamma superscript lambda] of Broken [lambda]-Extremals     79
The "Geometric" Index of a Critical Extremal g     85
Free Index Forms     93
Free Linear Differential and Boundary Conditions     93
Relative Minima under General End Conditions C[subscript r]     106
Focal Conditions and Sturm-Like Theorems
The Focal Point Theorem     117
Focal Conditions, Derived andFree     117
Conjugate Families (von Escherich)     124
Proof of Focal Point Theorem 17.3     128
Sturm-Like Theorems in E[subscript m+1]     140
The Separation Theorem in E[subscript m+1]     140
Comparison of Free Focal Conditions     144
The Sturm Comparisons When m = 1     155
The Variation of Conjugate Points     164
General Boundary Conditions
The Oscillation Number, Index W[subscript 0]     169
Conjugate Points and Characteristic Roots     169
The Auxiliary Theorem on Quadratic Forms     172
Proof of Oscillation Theorem 24.1     174
Quadratic End Forms of the System W[subscript r]     176
A Minimizing Periodic Extremal g     181
Selfadjoint Boundary Conditions     185
General Mixed Boundary Conditions     185
Conditions for Selfadjointness     188
The Reduction of Selfadjoint BC to Canonical Form     190
Prestructures for Characteristic Root Theory
Admissible Canonical Systems W[subscript r]     197
The Dependence on [lambda]     197
Canonical Systems W[subscript r]     202
The Existence of Characteristic Roots     209
Selfconjugate Limit Values      209
Comparison of Characteristic Roots     216
The W[subscript r]-Nuclear Comparison Test     219
Oscillation Criteria     224
Free Linear Conditions
Free Second-Order Differential Equations     229
Solution of the Differential Equations (1.5)     231
Conjugate Points     232
Second-Order DE with a Parameter [alpha]     236
The End Space ((W[subscript r])) of W[subscript r]     238
Subordinate Quadratic Forms and Their Complementary Forms
Auxiliary Theorem 25.1     241
The Case of a Nonsingular Form P(0,s)     243
Proof of Theorem 25.1 Completed     246
Model Families of Quadratic Forms
Model Families F     250
Model Families G     251
The Principal Theorem of Appendix III     253
References     255
Index     259