ISBN: 3-540-66243-X
TITLE: Fiscal Policy, Public Debt and the Term Structure of Interest Rates
AUTHOR: Demmel, Roland
TOC:

Introduction 1
Chapter 1. Classical Calculus of Variations 5
1. Euler Equation 5
1.1. Brachistochrone Problem 5
1.2. Euler Equation 6
1.3. Geodesics on a Riemannian Manifold 9
2. Hamiltonian Formalism 12
2.1. Legendre Transform 12
2.2. Canonical Variables 15
2.3. Mechanical Meaning of the Canonical Variables 16
2.4. Variation Formula for a Functional with Movable Endpoints 17
2.5. Transversality Conditions in the Problem with Movable Endpoints 18
2.6. WeierstrassErdmann Conditions 20
2.7. HamiltonJacobi Equation 23
3. Theory of the Second Variation 25
3.1. Problem of the Second Variation 25
3.2. Legendre Necessary Condition 26
3.3. The Associated Problem and the Definition of a Conjugate Point 28
3.4. Necessary Conditions for the Positive Semidefiniteness of 2J 29
4. Riccati Equation 30
4.1. Sufficient Conditions for the Positive Definiteness of 2J 30
5. Morse Index 35
6. Jacobi Envelope Theorem 38
7. Strong Minimum 42
7.1. Weierstrass Necessary Condition 42
8. PoincarCartan Integral Invariant 45
8.1. Exterior Differential Forms 45
8.2. PoincarCartan Integral Invariant 47
8.3. Legendre Manifolds 50
9. Fields of Extremals 53
9.1. Hilbert Invariant Integral 53
9.2. Embedding an Extremal in a Field and Focal Points 55
Chapter 2. Riccati Equation in the Classical Calculus of Variations 60
1. Riccati Equation as a Sufficient Condition for Positivity of the Second Variation 60
2. Riccati Equation for a Problem with Differential Constraints 62
2.1. Problem with Differential Constraints 63
2.2. Optimal Control Problem 63
2.3. Linear-Quadratic Problem 64
2.4. Bellman Equation 66
3. Riccati Equation and the Grassmann Manifold 68
3.1. Grassmann Manifold. 69
3.2. Riccati Equation as a Flow on the Grassmann Manifold 70
4. Grassmann Manifolds of Lower Dimension 72
4.1. Quaternions 73
4.2. Homotopic Paths 77
Chapter 3. Lie Groups and Lie Algebras 80
1. Lie Groups: Definition and Examples 80
2. Lie Algebras 84
2.1. Vector Fields on a Manifold 85
2.2. Lie Algebras 87
3. Lie Groups of Lower Dimension 88
3.1. Topological Structure of the Groups SO(3) and Spin(3) 88
3.2. Topological Structure of the Group SL(2, R) 90
3.3. Topological Structure of the Groups Sp(1, R), U(1), and SU(2) 90
4. Adjoint Representation and Killing Form 91
4.1. Adjoint Representation 91
4.2. Killing Form 93
4.3. Subalgebras and Ideals 93
5. Semisimple Lie Groups 96
5.1. Compact Lie Algebras 98
6. Homogeneous and Symmetrical Spaces 100
6.1. Symmetrical Spaces 101
7. Totally Geodesic Submanifolds 107
7.1. Lie Group Isometries 107
7.2. Geodesics in the Quotient Space of Lie Groups 110
Chapter 4. Grassmann Manifolds 112
1. Three Approaches to the Description of the Grassmann Manifolds 112
1.1. Local Coordinates on the Grassmann Manifold 112
1.2. Invariant Description of Grassmann Manifolds 114
1.3. Metric on the Grassmann Manifold 114
1.4. Grassmann Manifolds as Symmetrical Spaces 114
1.5. Plcker Embeddings 115
2. LagrangeGrassmann Manifolds 117
2.1. Coordinates on a LagrangeGrassmann Manifold 118
2.2. LagrangeGrassmann Manifold as a Homogeneous Space 119
2.3. The Manifold (R2n) as a Symmetrical Space 123
3. Riccati Equation as a Flow on the Manifold Gn(R2n) 123
4. Systems Associated with a Linear System of Differential Equations 126
4.1. Associated Systems on Grassmann Manifolds 128
Chapter 5. Matrix Double Ratio 130
1. Matrix Double Ratio on the Grassmann Manifold 130
2. Clifford Algebras 135
3. Totally Geodesic Submanifolds of Grassmann Manifolds 137
4. Curves with a Scalar Double Ratio 143
5. Fourth Harmonic as a Geodesic Symmetry 147
5.1. Manifold of Isotropic Planes 148
6. Clifford Parallels 150
7. Connection Between Clifford Parallels and Isoclinic Planes 154
8. Matrix Double Ratio on the LagrangeGrassmann Manifold 155
9. MorseMaslovArnol'd Index in the LerayKashivara Form 158
10.Fourth Harmonic as an Isometry of the LagrangeGrassmann Manifold 162
11.Application of the Matrix Double Ratio to the Study of the Riccati Equation 162
Chapter 6. Complex Riccati Equations 166
1. CartanSiegel Domains 166
2. KleinPoincar Upper Half-Plane and Generalized Siegel Upper Half-Plane 174
2.1. Generalized Siegel Upper Half-Plane 177
2.2. Siegel Half-Plane as a Symmetrical Space 178
2.3. Action of Sp(n, R) on the Boundary of the Siegel Half-Plane 184
2.4. Cayley Transform 185
3. Complexified Riccati Equation as a Flow on the Generalized Siegel Upper Half-Plane 187
4. Flow on CartanSiegel Homogeneity Domains 189
4.1. Riccati-Type Equation for a Linear System Whose Matrix Belongs to a Given Lie Algebra 190
4.2. Flow on the Siegel Homogeneity Domain of Type I 192
4.3. Flow on the Siegel Homogeneity Domain of Type II 194
4.4. Flow on the Siegel Homogeneity Domain of Type IV 196
5. Matrix Analog of the Schwarz Differential Operator 198
5.1. Classical Schwarz Differential Operator 200
5.2. Schwarz Operator and a Linear Second-Order Differential Equation 202
5.3. Schwarz Operator and the Riccati Equation 203
5.4. Matrix Analog of the Schwarz Operator 205
Chapter 7. Higher-Dimensional Calculus of Variations 208
1. Minimal Surfaces 208
2. Necessary Optimality Conditions for a Multiple Integral 212
2.1. Euler Equation 213
2.2. Second Variation 215
2.3. Variational Equation 216
3. Vector Bundles 217
4. Distributions and the Frobenius Theorem 219
5. Connection in a Linear Bundle 227
6. Levi-Civita Connection 230
6.1. Torsion and Curvature of a Connection of a Vector Bundle 233
7. Nonnegativity Conditions of the Second Variation 236
8. Field Theory in the Weyl Form 240
9. Caratheodory Transformation 244
9.1. Condition for Realizability of the Caratheodory Transformation 247
10.Field Theory in the Caratheodory Form 248
Chapter 8. On the Quadratic System of Partial Differential Equations Related to the Minimization Problem for a Multiple Integral 254
1. Riccati Equation in the Case of the Degenerate Legendre Condition 254
2. Reducing the Dirichlet Integral to the Integral of Its Principal Part 257
3. Relation of the Riccati Partial Differential Equation to the Euler Equation 261
3.1. Compactification of the Space on Which the Riccati Partial Differential Equation is Defined 263
4. Connection Defined by a Solution to the Riccati Partial Differential Equation 264
4.1. Potentiality Condition for Tensor Fields 270
Epilogue 272
Appendix to the English Edition 273
References 276
Index 282
END
