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Publisher:

Dover Publications

Publication Date:

1986

Number of Pages:

418

Format:

Paperback

Price:

19.95

ISBN:

978-0486650678

Category:

Monograph

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

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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 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 | |||||||

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 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 |

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