Motivated and an enjoyable read while maintaining rigor and has a lot of examples too. I would recommend it highly for self study.
Author(s): Bruce van Brunt
Series: Universitext
Edition: 1
Publisher: Springer
Year: 2003
Language: English
Pages: 295
Cover......Page 1
Title......Page 3
Preface......Page 6
Table of Contents......Page 9
1.1 Introduction......Page 12
1.2 The Catenary and Brachystochrone Problems......Page 14
1.3 Hamilton’s Principle......Page 21
1.4 Some Variational Problems from Geometry......Page 25
1.5 Optimal Harvest Strategy......Page 32
2.1 The Finite-Dimensional Case......Page 34
2.2 The Euler-Lagrange Equation......Page 39
2.3 Some Special Cases......Page 47
2.4 A Degenerate Case......Page 53
2.5 Invariance of the Euler-Lagrange Equation......Page 55
2.6 Existence of Solutions to the Boundary-Value Problem......Page 60
3.1 Functionals Containing Higher-Order Derivatives......Page 66
3.2 Several Dependent Variables......Page 71
3.3 Two Independent Variables......Page 76
3.4 The Inverse Problem......Page 81
4.1 The Finite-Dimensional Case and Lagrange Multipliers......Page 84
4.2 The Isoperimetric Problem......Page 94
4.3 Some Generalizations on the Isoperimetric Problem......Page 106
5.1 The Sturm-Liouville Problem......Page 113
5.2 The First Eigenvalue......Page 119
5.3 Higher Eigenvalues......Page 125
6.1 Holonomic Constraints......Page 129
6.2 Nonholonomic Constraints......Page 136
6.3 Nonholonomic Constraints in Mechanics......Page 142
7.1 Natural Boundary Conditions......Page 144
7.2 The General Case......Page 153
7.3 Transversality Conditions......Page 159
8.The Hamiltonian Formulation......Page 167
8.1 The Legendre Transformation......Page 168
8.2 Hamilton’s Equations......Page 172
8.3 Symplectic Maps......Page 179
8.4 The Hamilton-Jacobi Equation......Page 183
8.5 Separation of Variables......Page 193
9.1 Conservation Laws......Page 209
9.2 Variational Symmetries......Page 210
9.3 Noether’s Theorem......Page 215
9.4 Finding Variational Symmetries......Page 221
10.1 The Finite-Dimensional Case......Page 229
10.2 The Second Variation......Page 233
10.3 The Legendre Condition......Page 235
10.4 The Jacobi Necessary Condition......Page 240
10.5 A Sufficient Condition......Page 249
10.6 More on Conjugate Points......Page 253
10.7 Convex Integrands......Page 265
A.1 Taylor’s Theorem.......Page 269
A.2 The Implicit Function Theorem......Page 273
A.3 Theory of Ordinary Differential Equations......Page 276
B.1 Normed Spaces......Page 280
B.2 Banach and Hilbert Spaces......Page 285
References......Page 289
Index......Page 292
