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