4.6 One-Sided Variational Problems . . . . . . . . . . . . . . . . . . . . . . . 303
4.7 Introduction to the Famous Scientists . . . . . . . . . . . . . . . . . . . . 315
5 Variational Problems of Conditional Extrema . . . . . . . . . . . . . . . . 319
5.1 Variational Problems with Holonomic Constraints . . . . . . . . . . 319
5.2 Variational Problems with Differential Constraints . . . . . . . . . . 326
5.3 Isoperimetric Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
5.4 Extremal Problems of Mixed Type Functionals . . . . . . . . . . . . . 344
5.4.1 Extremal Problems of Simple Mixed Type
Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
5.4.2 Euler Equations of 2-D, 3-D and n-D Problems . . . . . . 353
5.5 Introduction to the Famous Scientists . . . . . . . . . . . . . . . . . . . . 359
6 Variational Problems in Parametric Forms . . . . . . . . . . . . . . . . . . . 365
6.1 Parametric Forms of Curves and Homogeneous Condition . . . . 365
6.2 Isoperimetric Problems in Parametric Forms and Geodesic
Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
6.3 Extrema of Functionals with Variable Boundaries
and Parametric Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
7 Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
7.1 Sets and Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
7.2 Sets and Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
7.3 Normal Orthogonal System and Fourier Series . . . . . . . . . . . . . 400
7.4 Operators and Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
7.5 Derivatives of Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
7.6 Variational Principles of Operator Equations . . . . . . . . . . . . . . 418
7.7 Variational Problems of Equivalence with Boundary Value
Problem of Self Conjugate Ordinary Differential Equation . . . . . 421
7.8 Variational Problems of Equivalence with Boundary Value
Problem of Self Conjugate Partial Differential Equation . . . . . . 428
7.9 The Friedrichs Inequality and Poincaré Inequality . . . . . . . . . . . 436
7.10 To the Famous Scientists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
8 Direct Methods of Variational Problems . . . . . . . . . . . . . . . . . . . . . 453
8.1 Minimizing (Maximizing) Sequence . . . . . . . . . . . . . . . . . . . . . 454
8.2 The Euler Finite Difference Method . . . . . . . . . . . . . . . . . . . . . 457
8.3 The Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
8.4 The Kantorovich Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
8.5 The Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
8.6 The Least Square Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
8.7 Eigenvalues and Eigenfunctions of Operator Equations . . . . . . . 487
8.8 Introduction to the Famous Scientists . . . . . . . . . . . . . . . . . . . . 503
Contents
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