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```I   Mathematical foundation                                     1
1 The foundations of calculus of variations                     3
1.1  The fundamental problem and lemma of calculus of variations  3
1.2  The Legendre test  ................... .....            7
1.3  The Euler-Lagrange differential equation . ...........  9
1.4  Application: Minimal path problems . .............     11
1.4.1  Shortest curve between two points . .......... 12
1.4.2  The brachistochrone problem  . .............    14
1.4.3  Fermat's principle  . ...................       18
1.4.4  Particle moving in the gravitational field ....... .  20
1.5  Open boundary variational problems . .............     21
2 Constrained variational problems                             25
2.1  Algebraic boundary conditions  . ................ 25
2.2  Lagrange's solution  . .......................         27
2.3  Application: Iso-perimetric problems . .............   29
2.3.1  Maximal area under curve with given length ...... 29
2.3.2  Optimal shape of curve of given length under gravity . 31
2.4  Closed-loop integrals ................... ....         35
3 Multivariate functionals                                     37
3.1  Functionals with several functions . ............ .  . 37
3.2  Variational problems in parametric form  . ...........  38
3.3  Functionals with two independent variables  . .........  39
3.4  Application: Minimal surfaces . .................      40
3.4.1  Minimal surfaces of revolution . .............  43
3.5  Functionals with three independent variables  . ........  44
4 Higher order derivatives                                     49
4.1  The Euler-Poisson equation  . ..................       49
4.2  The Euler-Poisson system of equations . ............ 51
4.3  Algebraic constraints on the derivative .... . . . ...... .52
4.4  Application: Linearization of second order problems  .... . .54
5 The inverse problem of the calculus of variations             57
5.1  The variational form of Poisson's equation . .......... 58
5.2  The variational form of eigenvalue problems . .........  59
5.2.1  Orthogonal eigensolutions . ............... 61
5.3  Application: Sturm-Liouville problems . ............ 62
5.3.1  Legendre's equation and polynomials . ......... 64
6 Direct methods of calculus of variations                      69
6.1  Euler's method  ..........................              69
6.2  Ritz method ........ .         ................         71
6.2.1  Application: Solution of Poisson's equation ...... 75
6.3  Galerkin's method  ........................ 76
6.4  Kantorovich's method  ............ .......... 78
II   Engineering applications                                  85
7 Differential geometry                                         87
7.1  The geodesic problem  . ..................... 87
7.1.1  Geodesics of a sphere . .................. 89
7.2  A system of differential equations for geodesic curves .... .  90
7.2.1  Geodesics of surfaces of revolution . .......... 92
7.3  Geodesic curvature . ....................... 95
7.3.1  Geodesic curvature of helix . .............. 97
7.4  Generalization of the geodesic concept . ............ 98
8 Computational geometry                                       101
8.1  Natural splines  ..........................           . 101
8.2  B-spline approximation ................... ... 104
8.3  B-splines with point constraints . . . . . .  .  .  .  .  .  .  .. . 109
8.4  B-splines with tangent constraints  . . . . . . . . . . . .  . 112
8.5  Generalization to higher dimensions  . . . . . . . . . . . . . 115
9 Analytic mechanics                                           119
9.1  Hamilton's principle for mechanical systems . ......... 119
9.2  Elastic string vibrations .     . . . . .  . . . . . .  . 120
9.3  The elastic membrane  .        . . . . . .  . . . . . .  . 125
9.3.1  Nonzero boundary conditions . ............. 130
9.4  Bending of a beam under its own weight . ........... 132
10 Computational mechanics                                     139
10.1 Three-dimensional elasticity .  . . . . .  . . . . . .  . 139
10.2 Lagrange's equations of motion  .  . . . . .  . . . . .  . 142
10.2.1 Hamilton's canonical equations . ............ .146
10.3 Heat conduction  .   ........................ 148
10.4 Fluid mechanics .  ........................           . 150
10.5 Computational techniques .        . . . . .  . . . . .  . 153
10.5.1  Discretization of continua  . . . . . . . .  .  .  .  .  .   .  153
10.5.2 Computation of basis functions . ............ 155
Closing Remarks                                                159
Notation                                                       161

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Library of Congress subject headings for this publication: Calculus of variations, Engineering mathematics