# The Variational Principles of Mechanics

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## Description

Analytical mechanics is, of course, a topic of perennial interest and usefulness in physics and engineering, a discipline that boasts not only many practical applications, but much inherent mathematical beauty. Unlike many standard textbooks on advanced mechanics, however, this present text eschews a primarily technical and formalistic treatment in favor of a fundamental, historical, philosophical approach. As the author remarks, there is a tremendous treasure of philosophical meaning behind the great theories of Euler and Lagrange, Hamilton, Jacobi, and other mathematical thinkers.

Well-written, authoritative, and scholarly, this classic treatise begins with an introduction to the variational principles of mechanics including the procedures of Euler, Lagrange, and Hamilton.

Ideal for a two-semester graduate course, the book includes a variety of problems, carefully chosen to familiarize the student with new concepts and to illuminate the general principles involved. Moreover, it offers excellent grounding for the student of mathematics, engineering, or physics who does not intend to specialize in mechanics, but wants a thorough grasp of the underlying principles.

The late Professor Lanczos (Dublin Institute of Advanced Studies) was a well-known physicist and educator who brought a superb pedagogical sense and profound grasp of the principles of mechanics to this work, now available for the first time in an inexpensive Dover paperback edition. His book will be welcomed by students, physicists, engineers, mathematicians, and anyone interested in a clear masterly exposition of this all-important discipline.

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Well-written, authoritative, and scholarly, this classic treatise begins with an introduction to the variational principles of mechanics including the procedures of Euler, Lagrange, and Hamilton.

Ideal for a two-semester graduate course, the book includes a variety of problems, carefully chosen to familiarize the student with new concepts and to illuminate the general principles involved. Moreover, it offers excellent grounding for the student of mathematics, engineering, or physics who does not intend to specialize in mechanics, but wants a thorough grasp of the underlying principles.

The late Professor Lanczos (Dublin Institute of Advanced Studies) was a well-known physicist and educator who brought a superb pedagogical sense and profound grasp of the principles of mechanics to this work, now available for the first time in an inexpensive Dover paperback edition. His book will be welcomed by students, physicists, engineers, mathematicians, and anyone interested in a clear masterly exposition of this all-important discipline.

show more

## Product details

- Paperback | 418 pages
- 137 x 215 x 22mm | 470g
- 22 Sep 1986
- Dover Publications Inc.
- New York, United States
- English
- New edition
- New edition
- Illustrations, unspecified
- 0486650677
- 9780486650678
- 86,814

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## Table of 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 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 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'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 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 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 equations

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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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 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'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 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 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 equations

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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## Review Text

Philosophic, less formalistic approach to perennially important field of analytical mechanics....

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