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 nonholonomic 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 lambdamethod 

6. Nonholonomic 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 deltaprocess 

10. The stationary value of a definite integral treated by the calculus of variations 

11. The EulerLagrange differential equations for n degrees of freedom 

12. Variation with auxiliary conditions 

13. Nonholonomic 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 lambdafactor 

9. Nonholonomic 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 HamiltonJacobi 

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 opticomechanical analogy 

8. The significance of Hamilton's partial differential equation in the theory of wave motion 

9. The geometrization of dynamics. NonRiemannian geometrics. The metrical significance of Hamilton's partial differential equation 
IX. Relativistic Mechanics 

1. Historical Introduction 

2. Relativistic kinematics 

3. Minkowski's fourdimensional 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 fourdimensional world 

11. The planetary orbits in Einstein's gravitational theory 

12. The gravitational bending of light rays 

13. The gravitational redshirt of the spectral lines 

Bibliography 
X. Historical Survey 
XI. Mechanics of the Continua 

1. The variation of volume integrals 

2. Vectoranalytic 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 energymomentum tensor 

23. The ten conservation laws 

24. The dynamic law in field theoretical derivation 
Appendix I; Appendix II; Bibliography; Index 