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BK

4th ed.

New York : Dover Publications, 1986

xxix, 418 s. : il. ; 22 cm

ISBN 0-486-65067-7 (brož.)

Dover books on physics and chemistry

Obsahuje rejstřík

000102907

CONTENTS // INTRODUCTION // SECTION PAGE // 1. The variational approach to mechanics xxi // 2. The procedure of Euler and Lagrange xx|j // 3. Hamilton’s procedure xxiü // 4. The calculus of variations xxiv // 5. Comparison between the vectorial and the variational // treatments of mechanics xxiv // 6. Mathematical evaluation of the variational principles xxv // 7. Philosophical evaluation of the variational approach // to mechanics xxvi // I. THE BASIC CONCEPTS OF ANALYTICAL MECHANICS // 1. The principal viewpoints of analytical mechanics 3 // 2. Generalized coordinates 6 // 3. The configuration space 12 // 4. Mapping of the space on itself 14 // 5. Kinetic energy and Riemannian geometry 17 // 6. Holonomic and non-holonomic mechanical systems 24 // 7. Work function and generalized force 27 // 8. Scleronomic and rheonomic systems. The law of the // conservation of energy 31 // II. THE CALCULUS OF VARIATIONS // 1. The general nature of extremum problems 35 // 2. The stationary value of a function 38 // 3. The second variation 40 // 4. Stationary value versus extremum value 42 // 5. Auxiliary conditions. The Lagrangian X-method 43 // 6. Non-holonomic auxiliary conditions 48 // 7. The stationary value of a definite integral 49 // 8. The fundamental processes of the calculus of variations 54 // 9. The commutative properties of the 5-process 56 // XV // XVi // Contents // SECTION PAGE // 10. The stationary value of a definite integral treated // by the calculus of variations 57

11. The Euler-Lagrange differential equations for n degrees // of freedom 60 // 12. Variation with auxiliary conditions 62 // 13. Non-holonomic conditions 65 // 14. Isoperimetric conditions 66 // 15. The calculus of variations and boundary conditions. // The problem of the elastic bar 68 // III. THE PRINCIPLE OF VIRTUAL WORK // t. The principle of virtual work for reversible displacements 74 // 2. The equilibrium of a rigid body 78 // 3. Equivalence of two systems of forces 79 // 4. Equilibrium problems with auxiliary conditions 80 // 5. Physical interpretation of the Lagrangian multiplier method 83 // 6. Fourier’s inequality 86 // IV. D’ALEMBERT’S PRINCIPLE // 1. The force of inertia 88 // 2. The place of d’Alembert’s principle in mechanics 92 // 3. The conservation of energy as a consequence of // d’Alembert’s principle 94 // 4. Apparent forces in an accelerated reference system. // Einstein’s equivalence hypothesis 96 // 5. Apparent forces in a rotating reference system 100 // 6. Dynamics of a rigid body. The motion of the centre of mass 103 // 7. Dynamics of a rigid body. Euler’s equations 104 // 8. Gauss’ principle of least restraint 106 // V. THE LAGRANGIAN EQUATIONS OF MOTION // 1. Hamilton’s principle 111 // 2. The Lagrangian equations of motion and their invariance // relative to point transformations 115 // 3. The energy theorem as a consequence of Hamilton’s // principle 119 // 4. Kinosthenic or ignorable variables and their elimination 125 // 5. The forceless

mechanics of Hertz 130 // 6. The time as kinosthenic variable; Jacobi’s principle; // the principle of least action 132 // Contents // xvii // SECTION PAGE // 7. Jacobi’s principle and Riemannian geometry 138 // g! Auxiliary conditions; the physical significance of the // Lagrangian X-factor 141 // 9. Non-holonomic auxiliary conditions and polygenic forces 146 // 10. Small vibrations about a state of equilibrium 147 // VI. THE CANONICAL EQUATIONS OF MOTION // 1. Legendre’s dual transformation 161 // 2. Legendre’s transformation applied to the // Lagrangian function 164 // 3. Transformation of the Lagrangian equations of motion 166 // 4. The canonical integral 168 // 5. The phase space and the space fluid 172 // 6. The energy theorem as a consequence of the // canonical equations 175 // 7. Liouville’s theorem 177 // 8. Integral invariants, Helmholtz1 circulation theorem 180 // 9. The elimination of ignorable variables 183 // 10. The parametric form of the canonical equations 185 // VII. CANONICAL TRANSFORMATIONS // 1. Coordinate transformations as a method of solving // mechanical problems 193 // 2. The Lagrangian point transformations 195 // 3. Mathieu’s and Lie’s transformations 201 // 4. The general canonical transformation 204 // 5. The bilinear differential form 207 // 6. The bracket expressions of Lagrange and Poisson 212 // 7. Infinitesimal canonical transformations 216 // 8. The motion of the phase fluid as a continuous succession // of canonical transformations 219

// 9. Hamilton’s principal function and the motion of the // phase fluid 222 // VIII. THE PARTIAL DIFFERENTIAL EQUATION OF HAMILTON-JACOBI // 1. The importance of the generating function for the // problem of motion 229 // 2. Jacobi’s transformation theory 231 // 3. Solution of the partial differential equation by separation 239 // XVlll // Contents // section page // 4. Delaunay’s treatment of separable periodic systems 243 // 5. The role of the partial differential equation in the theories // of Hamilton and Jacobi 254 // 6. Construction of Hamilton’s principal function with the // help of Jacobi’s complete solution 262 // 7. Geometrical solution of the partial differential equation. // Hamilton’s optico-mechanical analogy 264 // 8. The significance of Hamilton’s partial differential equation // in the theory of wave motion 276 // 9. The geometrization of dynamics. Non-Riemannian // geometries. The metrical significance of Hamilton’s // partial differential equation 280 // IX. RELATIVISTIC MECHANICS // 1. Historical introduction 291 // 2. Relativistic kinematics 294 // 3. Minkowski’s four-dimensional world 300 // 4. The Lorentz transformations 303 // 5. Mechanics of a particle 314 // 6. The Hamiltonian formulation of particle dynamics 319 // 7. The potential energy V 320 // 8. Relativistic formulation of Newton’s scalar theory of // gravitation 322 // 9. Motion of a charged particle 324 // 10. Geodesics of a four-dimensional world 329 // 11. The planetary orbits in Einstein’s

gravitational theory 330 // 12. The gravitational bending of light rays 336 // 13. The gravitational red-shift of the spectral lines 338 // Bibliography 340 // X. HISTORICAL SURVEY 341 // XL MECHANICS OF THE CONTINUA // 1. The variation of volume integrals 352 // 2. Vector-analytic tools 354 // 3. Integral theorems 357 // 4. The conservation of mass 35g // 5. Hydrodynamics of ideal fluids 359 // 6. The hydrodynamic equations in Lagrangian formulation 360 // 7. Hydrostatics 332 // 8. The circulation theorem 354 // 9. Euler’s form of the hydrodynamic equations 365 // Contents xix // SECTION PAGE // 10. The conservation of energy 368 // 11. Elasticity. Mathematical tools 369 // 12. The strain tensor 372 // 13. The stress tensor 374 // 14. Small elastic vibrations 375 // 15. The Hamiltonization of variational problems 376 // 16. Young’s modulus. Poisson’s ratio 378 // 17. Elastic stability 379 // 18. Electromagnetism. Mathematical tools 380 // 19. The Maxwell equations 381 // 20. Noether’s prindple 384 // 21. Transformation of the coordinates 386 // 22. The symmetric energy-momentum tensor 389 // 23. The ten conservation laws 393 // 24. The dynamical law in field theoretical derivation 394 // Appendix I 397 // Appendix II 401 // Bibliography 407 // Index 409