Contents
Preface to the second edition xi
Preface to the first edition xiii
Acknowledgments xv
About the author xvii
List of notations xix
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 TheLegendretest ........................ 7
1.3 The Euler-Lagrange differential equation . . . . . . . . . . . . 9
1.4 Application:minimalpathproblems .............. 11
1.4.1 Shortest curve between two points . . . . . . . . . . . 12
1.4.2 Thebrachistochroneproblem .............. 14
1.4.3 Fermatsprinciple .................... 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 Lagrangessolution ........................ 27
2.3 Application:iso-perimetricproblems .............. 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-loopintegrals ....................... 35
3 Multivariate functionals 37
3.1 Functionalswithseveralfunctions................ 37
3.2 Variational problems in parametric form . . . . . . . . . . . . 38
3.3 Functionals with two independent variables . . . . . . . . . . 39
3.4 Application:minimalsurfaces .................. 40
3.4.1 Minimalsurfacesofrevolution.............. 43
3.5 Functionals with three independent variables . . . . . . . . . 44
vii
viii Contents
4 Higher order derivatives 49
4.1 TheEuler-Poissonequation ................... 49
4.2 TheEuler-Poissonsystemofequations ............. 51
4.3 Algebraicconstraintsonthederivative ............. 52
4.4 Linearization of second order problems . . . . . . . . . . . . . 54
5 The inverse problem of calculus of variations 57
5.1 The variational form of Poisson’s equation . . . . . . . . . . . 58
5.2 Thevariationalformofeigenvalueproblems .......... 59
5.2.1 Orthogonal eigensolutions . . . . . . . . . . . . . . . . 61
5.3 Sturm-Liouville problems . . . . . . . . . . . . . . . . . . . . 62
5.3.1 Legendre’s equation and polynomials . . . . . . . . . . 64
6 Analytic solutions of variational problems 69
6.1 Laplacetransformsolution .................... 69
6.2 Separationofvariables ...................... 71
6.3 Completeintegralsolutions ................... 76
6.4 Poissonsintegralformula .................... 80
6.5 Methodofgradients ....................... 85
7 Numerical methods of calculus of variations 89
7.1 Eulersmethod .......................... 89
7.2 Ritzmethod ............................ 91
7.2.1 Application: solution of Poisson’s equation . . . . . . 95
7.3 Galerkinsmethod ........................ 96
7.4 Kantorovichsmethod ...................... 98
7.5 Boundary integral method . . . . . . . . . . . . . . . . . . . . 103
II Engineering applications 109
8 Differential geometry 111
8.1 Thegeodesicproblem ......................111
8.1.1 Geodesicsofasphere...................113
8.2 A system of differential equations for geodesic curves . . . . . 114
8.2.1 Geodesics of surfaces of revolution . . . . . . . . . . . 116
8.3 Geodesiccurvature ........................119
8.3.1 Geodesiccurvatureofhelix ...............121
8.4 Generalization of the geodesic concept . . . . . . . . . . . . . 122
9 Computational geometry 125
9.1 Naturalsplines ..........................125
9.2 B-splineapproximation......................128
9.3 B-splineswithpointconstraints .................133
9.4 B-splineswithtangentconstraints ...............136
9.5 Generalizationtohigherdimensions ..............139
Contents ix
10 Variational equations of motion 143
10.1 Legendresdualtransformation .................143
10.2 Hamilton’s principle for mechanical systems . . . . . . . . . . 144
10.2.1 Newtonslawofmotion .................145
10.3 Lagrange’s equations of motion . . . . . . . . . . . . . . . . . 146
10.4 Hamiltonscanonicalequations .................147
10.4.1 Conservationofenergy..................149
10.5 Orbitalmotion ..........................150
10.6 Variational foundation of fluid motion . . . . . . . . . . . . . 153
11 Analytic mechanics 157
11.1 Elasticstringvibrations .....................157
11.2 Theelasticmembrane ......................162
11.2.1 Circularmembranevibrations..............165
11.2.2 Non-zero boundary conditions . . . . . . . . . . . . . . 167
11.3 Bending of a beam under its own weight . . . . . . . . . . . . 169
12 Computational mechanics 177
12.1 Three-dimensionalelasticity ...................177
12.2 Lagrangianformulation .....................180
12.3 Heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . 184
12.4 Fluidmechanics ..........................186
12.5 Theniteelementmethod ....................189
12.5.1 Finiteelementmeshing..................189
12.5.2 Shapefunctions......................191
12.5.3 Elementmatrixgeneration................195
12.5.4 Element matrix assembly and solution . . . . . . . . . 199
Closing remarks 205
References 207
Index 209
List of Figures 211
List of Tables 213

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