The Variational Principles of Mechanics

Title Page
Copyright Page
Dedication
Preface
	Preface to the Second Edition
	Preface to the Second Edition
	Preface to the Second Edition
Contents
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 approachto mechanics
The Variational Principles of Mechanics
I. The Basic Concepts of Analytical Mechanics
	1. The principal view points 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 theconservation 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 λ-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 5-process
	10. The stationary value of a definite integral treated by the calculus of variations
	11. The Euler-Lagrange differential equations for n degreesof 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’sprinciple
	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 λ-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 geometries. 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-shift 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 dynamical law in field theoretical derivation
Appendix I
Appendix II
Bibliography
Index