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The Variational Principles of Mechanics

Cornelius Lanczos
Dover Publications
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  1. The variational approach to mechanics
  2. The procedure of Euler and Lagrange
  3. Hamilton's procedure
  4. The calculus of variations
  5. Comparison between the vectorial and the variational treatments of mechanics
  6. Mathematical evaluation of the variational principles
  7. Philosophical evaluation of the variational approach to mechanics
I. The Basic Concepts of Analytical Mechanics
  1. The Principal viewpoints of analytical mechanics
  2. Generalized coordinates
  3. The configuration space
  4. Mapping of the space on itself
  5. Kinetic energy and Riemannian geometry
  6. Holonomic and non-holonomic mechanical systems
  7. Work function and generalized force
  8. Scleronomic and rheonomic systems. The law of the conservation of energy
II. The Calculus of Variations
  1. The general nature of extremum problems
  2. The stationary value of a function
  3. The second variation
  4. Stationary value versus extremum value
  5. Auxiliary conditions. The Lagrangian lambda-method
  6. Non-holonomic auxiliary conditions
  7. The stationary value of a definite integral
  8. The fundamental processes of the calculus of variations
  9. The commutative properties of the delta-process
  10. The stationary value of a definite integral treated by the calculus of variations
  11. The Euler-Lagrange differential equations for n degrees of freedom
  12. Variation with auxiliary conditions
  13. Non-holonomic conditions
  14. Isoperimetric conditions
  15. The calculus of variations and boundary conditions. The problem of the elastic bar
III. The principle of virtual work
  1. The principle of virtual work for reversible displacements
  2. The equilibrium of a rigid body
  3. Equivalence of two systems of forces
  4. Equilibrium problems with auxiliary conditions
  5. Physical interpretation of the Lagrangian multiplier method
  6. Fourier's inequality
IV. D'Alembert's principle
  1. The force of inertia
  2. The place of d'Alembert's principle in mech
  3. The conservation of energy as a consequence of d'Alembert's principle
  4. Apparent forces in an accelerated reference system. Einstein's equivalence hypothesis
  5. Apparent forces in a rotating reference system
  6. Dynamics of a rigid body. The motion of the centre of mass
  7. Dynamics of a rigid body. Euler's equations
  8. Gauss' principle of least restraint
V. The Lagrangian equations of motion
  1. Hamilton's principle
  2. The Lagrangian equations of motion and their invariance relative to point transformations
  3. The energy theorem as a consequence of Hamilton's principle
  4. Kinosthenic or ignorable variables and their elimination
  5. The forceless mechanics of Hertz
  6. The time as kinosthenic variable; Jacobi's principle; the principle of least action
  7. Jacobi's principle and Riemannian geometry
  8. Auxiliary conditions; the physical significance of the Lagrangian lambda-factor
  9. Non-holonomic auxiliary conditions and polygenic forces
  10. Small vibrations about a state of equilibrium
VI. The Canonical Equations of motion
  1. Legendre's dual transformation
  2. Legendre's transformation applied to the Lagrangian function
  3. Transformation of the Lagrangian equations of motion
  4. The canonical integral
  5. The phase space and the space fluid
  6. The energy theorem as a consequence of the canonical equations
  7. Liouville's theorem
  8. Integral invariants, Helmholtz' circulation theorem
  9. The elimination of ignorable variables
  10. The parametric form of the canonical equations
VII. Canonical Transformations
  1. Coordinate transformations as a method of solving mechanical problems
  2. The Lagrangian point transformations
  3. Mathieu's and Lie's transformations
  4. The general canonical transformation
  5. The bilinear differential form
  6. The bracket expressions of Lagrange and Poisson
  7. Infinitesimal canonical transformations
  8. The motion of the phase fluid as a continuous succession of canonical transformations
  9. Hamilton's principal function and the motion of the phase fluid
VIII. The Partial differential equation of Hamilton-Jacobi
  1. The importance of the generating function for the problem of m
  2. Jacobi's transformation theory
  3. Solution of the partial differential equation by separation
  4. Delaunay's treatment of separable periodic systems
  5. The role of the partial differential equation in the theories of Hamilton and Jacobi
  6. Construction of Hamilton's principal function with the help of Jacobi's complete solution
  7. Geometrical solution of the partial differential equation. Hamilton's optico-mechanical analogy
  8. The significance of Hamilton's partial differential equation in the theory of wave motion
  9. The geometrization of dynamics. Non-Riemannian geometrics. The metrical significance of Hamilton's partial differential equation
IX. Relativistic Mechanics
  1. Historical Introduction
  2. Relativistic kinematics
  3. Minkowski's four-dimensional world
  4. The Lorentz transformations
  5. Mechanics of a particle
  6. The Hamiltonian formulation of particle dynamics
  7. The potential energy V
  8. Relativistic formulation of Newton's scalar theory of gravitation
  9. Motion of a charged particle
  10. Geodesics of a four-dimensional world
  11. The planetary orbits in Einstein's gravitational theory
  12. The gravitational bending of light rays
  13. The gravitational red-shirt of the spectral lines
X. Historical Survey
XI. Mechanics of the Continua
  1. The variation of volume integrals
  2. Vector-analytic tools
  3. Integral theorems
  4. The conservation of mass
  5. Hydrodynamics of ideal fluids
  6. The hydrodynamic equations in Lagrangian formulation
  7. Hydrostatics
  8. The circulation theorem
  9. Euler's form of the hydrodynamic equations
  10. The conservation of energy
  11. Elasticity. Mathematical tools
  12. The strain tensor
  13. The stress tensor
  14. Small elastic vibrations
  15. The Hamiltonization of variational problems
  16. Young's modulus. Poisson's ratio
  17. Elastic stability
  18. Electromagnetism. Mathematical tools
  19. The Maxwell equa
  20. Noether's principle
  21. Transformation of the coordinates
  22. The symmetric energy-momentum tensor
  23. The ten conservation laws
  24. The dynamic law in field theoretical derivation
  Appendix I; Appendix II; Bibliography; Index