Introduction |

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 mechanics |

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 prin |

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 motion |

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 |

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 |

Bibliography |

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 equations |

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 |