9 Variational Principles in Mechanics and Their Applications . . . . . . 511
9.1 Fundamental Conceptions in Mechanics . . . . . . . . . . . . . . . . . . 512
9.1.1 System of Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 512
9.1.2 Constraints and Their Classification . . . . . . . . . . . . . . . 513
9.1.3 Actual Displacement and Virtual Displacement . . . . . . 514
9.1.4 Relations Between Strains and Displacements . . . . . . . 515
9.1.5 Work and Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
9.2 Principle of Virtual Displacement . . . . . . . . . . . . . . . . . . . . . . 524
9.2.1 Principle of Virtual Displacement for System
of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
9.2.2 Principle of Generalized Virtual Displacement
for Elastic Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
9.2.3 Principle of Generalized Virtual Displacement
for Elastic Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
9.3 Principle of the Minimum Potential Energy . . . . . . . . . . . . . . . 534
9.4 Principle of Complementary Virtual Work . . . . . . . . . . . . . . . . 539
9.5 Principle of the Minimum Complementary Energy . . . . . . . . . . 543
9.6 The Hamilton Principles and Their Applications . . . . . . . . . . . . 544
9.6.1 The Hamilton Principle of System of Particles . . . . . . . 544
9.6.2 The Hamilton Principle of Elastic Body . . . . . . . . . . . . 562
9.7 The Hamilton’s Canonical Equations . . . . . . . . . . . . . . . . . . . . 578
9.8 The Hellinger-Reissner Generalized Variational Principles . . . . . 585
9.9 The Hu Haichang-Kyuichiro Washizu Generalized
Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
9.10 The Maupertuis-Lagrange Principle of Least Action . . . . . . . . . 592
9.11 Introduction to the Famous Scientists . . . . . . . . . . . . . . . . . . . . 598
10 Variational Problems of Functionals with Vector,
Tensor and Hamiltonian Operators . . . . . . . . . . . . . . . . . . . . . . . . 609
10.1 Basic Properties of the Tensor Inner Product Operations
and Fundamental Lemma of the Variation of Functional
with Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610
10.2 The Euler Equations of Functionals with Vector, Modulus
of Vector and Hamiltonian Operators . . . . . . . . . . . . . . . . . . . . 617
10.3 The Euler Equations of Gradient Type Functionals . . . . . . . . . . 638
10.4 The Euler Equations of Divergence Type Functionals . . . . . . . . 652
10.5 The Euler Equations of Rotation Type Functionals . . . . . . . . . . 669
10.6 Variational Problems of Functionals with Parallel-Type
Inner Product Tensors and Hamiltonian Operators . . . . . . . . . . 685
10.6.1 Variational Formula Derivations of Gradients,
Divergences and Rotations of Parallel-Type Inner
Product Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
10.6.2 The Euler Equations and Natural Boundary
Conditions of the Functionals with Parallel-Type
Inner Product Tensors and Hamiltonian Operators . . . . 690
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