2 Variational Problems with Fixed Boundaries . . . . . . . . . . . . . . . . . 85
2.1 Examples of the Classical Variational Problems . . . . . . . . . . . . 86
2.2 Fundamental Conceptions of the Calculus of Variations . . . . . . 90
2.3 Variations of the Simplest Functionals and Necessary
Conditions of Extrema of Functionals . . . . . . . . . . . . . . . . . . . 101
2.4 The Euler Equations of the Simplest Functional . . . . . . . . . . . . 113
2.5 Several Special Cases of the Euler Equation and Their
Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2.6 Variational Problems Depending on Several Functions
of One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
2.7 Variational Problems Depending on Higher Order
Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
2.8 Variational Problems Depending on Functions of Several
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
2.9 Variational Problems of Complete Function . . . . . . . . . . . . . . . 172
2.10 Invariance of the Euler Equation . . . . . . . . . . . . . . . . . . . . . . . 179
2.11 Introduction to the Famous Scientists . . . . . . . . . . . . . . . . . . . . 188
3 Sufficient Conditions of Extrema of Functionals . . . . . . . . . . . . . . . 201
3.1 Extremal Curve Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
3.2 The Jacobi Conditions and Jacobi Equation . . . . . . . . . . . . . . . 204
3.3 The Weierstrass Functions and Weierstrass Conditions . . . . . . . 211
3.4 The Legendre Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
3.5 Sufficient Conditions of Extrema of Functionals . . . . . . . . . . . . 218
3.5.1 The Weierstrass Sufficient Conditions . . . . . . . . . . . . . 218
3.5.2 The Legendre Sufficient Conditions . . . . . . . . . . . . . . . 223
3.6 Higher Order Variations of Functionals . . . . . . . . . . . . . . . . . . 228
3.7 Introduction to the Famous Scientists . . . . . . . . . . . . . . . . . . . . 235
4 Problems with Variable Boundaries . . . . . . . . . . . . . . . . . . . . . . . . 239
4.1 Variational Problems of the Simplest Functional . . . . . . . . . . . . 239
4.2 Variational Problems of Functionals with Several
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
4.3 Variational Problems of Functionals with Higher Order
Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
4.3.1 Cases of Functionals with One Unknown Function
and Its Second Derivative . . . . . . . . . . . . . . . . . . . . . . 270
4.3.2 Cases of Functionals with One Unknown Function
and Its Several Order Derivatives . . . . . . . . . . . . . . . . 275
4.3.3 Cases of Functionals with Several Unknown
Functions and Their Several Order Derivatives . . . . . . . 281
4.4 Variational Problems of Functionals with Functions
of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
4.5 Extremal Curves with Cuspidal Points . . . . . . . . . . . . . . . . . . . 295
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