Calculus of Variations

Contents......Page 3
Introduction ......Page 6
1. The Euler Equation ......Page 15
2. Generalizations of the Euler Equation ......Page 23
3. Natural Boundary Conditions ......Page 30
4. Degenerate Euler Equation ......Page 35
5. Isoperimetric Problems ......Page 38
6. Parametric Form of the Theory ......Page 47
7. Invariance of the Euler Equation ......Page 53
8. The Legendre i:onditlon ......Page 56
1. The Legendre Transformation ......Page 59
2. The Distance Function - Reduction to Canonical Form ......Page 61
3. The Hamilton-Jacobi Partial Differential Equation ......Page 65
4. The Two Body Problem ......Page 79
5. The Homogeneous Case - Geodesics ......Page 81
6. Sufficient Conditions ......Page 87
7. Construction of a Field - The Conjugate Point ......Page 92
Supplementary Notes to Chapter II......Page 0
1. The Hamilton-Jacobi Equation ......Page 95
2. Correspondence between Solutions of the Hamilton-Jacobi Equation and Fields ......Page 99
3. Application to Differential Geometry ......Page 101
4. Analytical Representation of a Field ......Page 102
5. Conjugate Points ......Page 105
6. Application to Sturm's Theory ......Page 110
Introduction ......Page 114
Compactness in Function Space, Arzela's Theorem and Applications ......Page 119
Application to Geodesios: Lipschitz's Condition ......Page 123
Direct Variational Methods in the Theory of Integral Equations ......Page 126
Explicit Expression of Dirichlet's Integral for a Circle. Hadamard's Objection ......Page 129
Lower Semi-Continuity of Dirichlet's Integral for Harmonic Functions ......Page 131
Proof of Dirichlet's Principle for the Circle ......Page 132
"Distance" in Function Space. Triangle Inequalities ......Page 134
Construction of a Harmonic Function u by a "Smoothing Process" ......Page 137
Convergence of w_n ......Page 142
Proof that D(u)=d ......Page 144
Proof that the Function u Attains the Prescribed Boundary Values ......Page 145
Alternative Proof of Dirichlet's Principle ......Page 148
The Ritz Method ......Page 155
Method of Finite Differences ......Page 158
Existence and Uniqueness of the Solution ......Page 161
Practical Methods ......Page 162
Convergence of the Difference Equation to a Differential Equation ......Page 163
Method of Gradients ......Page 164
Extremum Properties of Eigenvalues ......Page 167
The Maximum-Minimum Property of the Eigenvalues ......Page 171
1. References ......Page 174
2. Notes on the Brachistochrone problem ......Page 175
3. The road of quickest ascent to the top of a mountain ......Page 176
4. The shortest connection between two points in a closed simply-connected region ......Page 179
5. The shortest connection in a plane between two points ......Page 183
6. Problems for which no solutions exist ......Page 185
7. Semi-continuity ......Page 193
8. Generalization of the Fundamental Lemma ......Page 194
10. Characterization of the longer great circle are ......Page 196
11. Integration of Euler's equation in special cases ......Page 197
12. The shortest connection between two points on a sphere ......Page 201
13. Application of Euler's equation to classical problems ......Page 208
14.. Invariance and the Euler expression ......Page 218
15. Transformation theory ......Page 252
16. An approximation theorem for minimum problems with side conditions ......Page 275