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Tables of Contents for The Geometry of Ordinary Variational Equations
Chapter/Section Title
Page #
Page Count
Chapter 1. INTRODUCTION
1
19
Chapter 2. BASIC GEOMETRIC TOOLS
20
21
2.1. Introduction
20
1
2.2. Distributions
21
4
2.3. Closed two-forms
25
4
2.4. Jet prolongations of fibered manifolds
29
2
2.5. Projectable vector fields
31
1
2.6. Calculus of horizontal and contact forms on fibered manifolds
32
5
2.7. Jet fields, connections, semispray connections and generalized connection on fibered manifolds
37
4
Chapter 3. LAGRANGEAN DYNAMICS ON FIBERED MANIFOLDS
41
11
3.1. Introduction
41
1
3.2. Lepagean one-forms and the first variation
41
8
3.3. Extremals of a Lagrangian
49
3
Chapter 4. VARIATIONAL EQUATIONS
52
28
4.1. Introduction
52
1
4.2. Locally variational forms
53
2
4.3. Lepagean two-forms, Lagrangean systems
55
9
4.4. A few words on global properties of the Euler-Lagrange mapping
64
2
4.5. Canonical form of Lepagean two-form and minimal-order Langrangians
66
7
4.6. Lower-order Lagrangean systems
73
2
4.7. Transformation properties of Lepagean forms
75
2
4.8. Holonomic constraints
77
3
Chapter 5. HAMILTONIAN SYSTEMS
80
17
5.1. Introduction
80
3
5.2. Hamilton two-form and generalized Hamilton equations
83
5
5.3. The characteristic and the Euler-Lagrange distributions
88
3
5.4. Structure theorems
91
4
5.5. The integration problem for variational ODE
95
2
Chapter 6. REGULAR LAGRANGEAN SYSTEMS
97
32
6.1. Introduction
97
1
6.2. Regularity as a geometrical concept
98
2
6.3. Regularity conditions for Lagrangians
100
3
6.4. Legendre transformation
103
4
6.5. Legendre chart expressions
107
3
6.6. Examples
110
6
6.7. Equivalence of dynamical forms and the inverse problem of the calculus of variations
116
13
Chapter 7. SINGULAR LAGRANGEAN SYSTEMS
129
20
7.1. Introduction
129
1
7.2. Classification of singular Lagrangean systems
130
2
7.3. The constraint algorithm for the Euler-Lagrange distribution
132
2
7.4. Applications of the constraint algorithm
134
7
7.5. Semiregular Lagrangean systems
141
2
7.6. Weakly regular Lagrangean systems
143
1
7.7. Generalized Legendre transformations
144
5
Chapter 8. SYMMETRIES OF LAGRANGEAN SYSTEMS
149
25
8.1. Introduction
149
1
8.2. Classification of symmetries, conserved functions
150
5
8.3. Point symmetries associated with Lagrangean systems
155
2
8.4. Point symmetries and first integrals
157
4
8.5. Applications of Noether equation and of Noether-Bessel-Hagen equation
161
10
8.6. Dynamical symmetries
171
3
Chapter 9. GEOMETRIC INTEGRATION METHODS
174
34
9.1. Introduction
174
1
9.2. The Liouville integration method
175
7
9.3. Jacobi complete integrals and the Hamilton-Jacobi integration method
182
5
9.4. Canonical transformations
187
3
9.5. Fields of extremals and the generalized Van Hove Theorem
190
5
9.6. Hamilton-Jacobi distributions for regular odd-order Lagrangean systems
195
3
9.7. Geodesic distance in a field of extremals
198
4
9.8. Two illustrative examples
202
5
9.9. A few remarks on integration of non-variational equations
207
1
Chapter 10. LAGRANGEAN SYSTEMS ON PI : R x M XXX R
208
21
10.1. Introduction
208
1
10.2. Lagrangean systems on R x M XXX R
208
2
10.3. Autonomous Lagrangean systems: Higher-order symplectic and presymplectic systems
210
10
10.4. Metric structures associated with regular first-order Lagrangean systems
220
9
Bibliography
229
17
Index
246