# Book:Charles Fox/An Introduction to the Calculus of Variations/Second Edition

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## Charles Fox:

## Charles Fox: *An Introduction to the Calculus of Variations (2nd Edition)*

Published $\text {1963}$, **Dover Publications**

- ISBN 0-486-65499-0

### Subject Matter

### Contents

- PREFACE

- $\text {I}$. THE FIRST VARIATION
- 1.1 Introduction
- 1.2 Ordinary maximum and minimum theory
- 1.3 Weak variations
- 1.4 The Eulerian characteristic equation
- 1.5 The Legendre test
- 1.6 Illustrations of the theory
- 1.7 Applications to statical problems; the catenary
- 1.8 Applications to dynamical problems
- 1.9 Applications to optical problems, paths of minimum time
- 1.10 Geodesics on a sphere
- 1.11 Brachistochrone
- 1.12 Minimal surfaces
- 1.13 Principle of least action. Inverse square law
- 1.14 Principle of least action. Direct distance law
- 1.15 A problem in fluid motion
- 1.16 Newton's solid of minimum resistance
- 1.17 Discontinuous solutions
- 1.18 Characteristic equation an identity

- $\text {II}$. THE SECOND VARIATION
- 2.1 Introduction
- 2.2 The second variation
- 2.3 Lemma $1$
- 2.4 Lemma $2$: Jacobi's accessory equation
- 2.5 Simple criteria for maxima and minima of $I$. The Legendre test
- 2.6 Conjugate points (kinetic foci)
- 2.7 Case when $B$ does not lie between $A$ and its nearest conjugate
- 2.8 The accessory equation
- 2.9 A property of conjugate points
- 2.10 Principle of least action
- 2.11 The catenary
- 2.12 Analytical methods for finding conjugate points
- 2.13 Conjugate points on the catenary
- 2.14 Conjugate points on a parabolic trajectory
- 2.15 Geodesics on spheres
- 2.16 Orbits under inverse square law of attraction
- 2.17 Orbit of a particle attracted by a force $m \mu r$
- 2.18 Properties of solutions of the accessory equation
- 2.19 Summary of the main results of Chapters $\text {I}$ and $\text {II}$

- $\text {III}$. GENERALIZATIONS OF THE RESULTS OF THE PREVIOUS CHAPTER
- 3.1 Introduction
- 3.2 Maxima and minima of integrals of the type $\ds I = \int_{t_0}^{t_1} \map F {q_1, q_2, \dotsc, q_n; \dot q_1, \dot q_2, \dotsc, \dot q_n; t} \rd t$
- 3.3 The second variation for integral $(1)$, $\S 3.2$
- 3.4 Conjugate points (kinetic foci) for integral $(1)$, $\S 3.2$
- 3.5 Integrals of the type $\int \map F {x, y, y_1, y_2, \dotsc, y_n} \rd x$, where $y_m = \d^m y / \d x^m$
- 3.6 The case of several independent variables and one dependent variable
- 3.7 Lemma on double integration
- 3.8 The characteristic equation for the integral $(1)$, $\S 3.6$
- 3.9 The second variation of integral $(1)$, $\S 3.6$
- 3.10 Application to physical and other problems
- 3.11 Application to theory of minimal surfaces

- $\text {IV}$. RELATIVE MAXIMA AND MINIMA AND ISOPERIMETRICAL PROBLEMS
- 4.1 Introduction
- 4.2 Relative maxima and minima
- 4.3 Examples illustrating theorem $11$
- 4.4 Examples $2$ and $3$
- 4.5 Example $4$
- 4.6 Further isoperimetric problems
- 4.7 Example $5$
- 4.8 Subsidiary equations of non-integral type
- 4.9 Example $6$. Geodesics
- 4.10 Examples $7$-$9$. Geodesics on a sphere
- 4.11 Non-holonomic dynamical constraints
- 4.12 The second variation
- 4.13 Isoperimetrical problems (second variation)
- 4.14 Subsidiary equations of non-integral type

- $\text {V}$. HAMILTON'S PRINCIPLE AND THE PRINCIPLE OF LEAST ACTION
- 5.1 Introduction
- 5.2 Degrees of freedom
- 5.3 Holonomic and non-holonomic systems
- 5.4 Conservative and non-conservative systems of force
- 5.5 Statement of Hamilton's principle
- 5.6 Statement of principle of least action
- 5.7 Proof of Hamilton's principle: preliminary remarks
- 5.8 First proof of Hamilton's principle for conservative holonomic systems
- 5.9 Second proof of Hamilton's principle for conservative holonomic systems
- 5.10 First proof of Hamilton's principle for non-holonomic systems
- 5.11 Second proof of Hamilton's principle for non-holonomic systems
- 5.12 Proof of Hamilton's principle for non-conservative dynamical systems
- 5.13 Proof of Lagrange's equations of motion
- 5.14 The energy equation for conservative fields of force
- 5.15 The second variation
- 5.16 A special variation of the extremals
- 5.17 Conjugate points
- 5.18 Positive semi-definite quadratic forms
- 5.19 A particle under no forces describes a geodesic
- 5.20 Dynamical paths related to geodesics on hypersurfaces
- 5.21 Hamilton's equations
- 5.22 The non-dynamical case when $L$ is homogeneous and of degree one in $\dot q_i$ $\paren {i = 1, 2, \dotsc, n}$
- 5.23 Path of minimum time in a stream with given flow

- $\text {VI}$. HAMILTON'S PRINCIPLE IN THE SPECIAL THEORY OF RELATIVITY
- 6.1 Introduction
- 6.2 The physical bases of the special theory of relativity
- 6.3 The Michelson and Morley experiment
- 6.4 The Trouton and Noble experiment
- 6.5 The principle of special relativity
- 6.6 Galilean and Newtonian conceptions of time
- 6.7 The transformations of the special theory of relativity
- 6.8 Relativity transformations for small time intervals
- 6.9 The space-time continuum
- 6.10 An approach to relativity dynamics of a particle
- 6.11 Applicability of Hamilton's principle to relativity mechanics
- 6.12 Equations of motion of a particle in relativity mechanics
- 6.13 Mass in relativity mechanics
- 6.14 Energy in relativity mechanics
- 6.15 Further observations

- $\text {VII}$. APPROXIMATION METHODS WITH APPLICATIONS TO PROBLEMS OF ELASTICITY
- 7.1 Introduction
- 7.2 Illustration using Euler's equation
- 7.3 Illustration using the Rayleigh-Ritz method
- 7.4 Rayleigh's method
- 7.5 The Rayleigh-Ritz method
- 7.6 Sturm-Liouville functions
- 7.7 The case of several independent variables
- 7.8 The specification of strain
- 7.9 The specification of stress
- 7.10 Conditions for equilibrium
- 7.11 Stress strain relations
- 7.12 The Saint-Venant torsion problem
- 7.13 The variational form of Saint-Venant's torsion problem
- 7.14 The torsion of beams with rectangular cross-section
- 7.15 Upper bounds for the integral $J$, $(9)$, $\S 7.13$
- 7.16 Lower bounds for the integral $J$, $(9)$, $\S 7.13$
- 7.17 Applications of the Trefftz method
- 7.18 Galerkin's method
- 7.19 Variations of the Rayleigh-Ritz and Galerkin methods

- $\text {VIII}$. INTEGRALS WITH VARIABLE END POINTS. HILBERT'S INTEGRAL
- 8.1 Introduction
- 8.2 First variation with one end point variable
- 8.3 First variation of an integral with both end points variable
- 8.4 Illustrations of the theory
- 8.5 The Brachistochrone
- 8.6 The second variation
- 8.7 The accessory equation
- 8.8 Focal points
- 8.9 The determination of focal points, (i) geometrical
- 8.10 The determination of focal points, (ii) analytical
- 8.11 Hilbert's integral
- 8.12 Fields of extremals
- 8.13 Hilbert's integral independent of the path of integration
- 8.14 The method of Carathéodory
- 8.15 The Bliss condition

- $\text {IX}$. STRONG VARIATIONS AND THE WEIERSTRASSIAN $E$ FUNCTION
- 9.1 Introduction
- 9.2 The Weierstrassian $E$ function in the simplest case
- 9.3 The simplified form of the Weierstrassian condition
- 9.4 The Weierstrassian condition by an alternative method
- 9.5 Conjugate points related to fields of extremals
- 9.6 Conditions for a strong maximum or minimum
- 9.7 Strong variations for integrals with two dependent variables
- 9.8 The Weierstrassian theory for integrals in parametric form
- 9.9 The Eulerian equation for $\ds \int_{t_1}^{t_2} \map G {x, y, \dot x, \dot y} \rd t$
- 9.10 The Weierstrassian $E$ function for $\ds \int_A^B \map G {x, y, \dot x, \dot y} \rd t$
- 9.11 Alternative forms for the $E$ function
- 9.12 Conditions for maxima and minima of $I = \ds \int_A^B \map G {x, y, \dot x, \dot y} \rd t$
- 9.13 Applications to special cases
- 9.14 Applications to geodesics on surfaces

- INDEX

**References.**The equations in each section are numbered from $(1)$ onwards. An equation in the same section as the point of reference is referred to by its number only; one in another section by its number and section number.

## Further Editions

## Errata

### Second Derivative at Maximum is Negative

Chapter $\text I$. The First Variation: $1.2$. Ordinary maximum and minimum theory:

*... it follows that at a maximum $\map {f''} a$ is negative and ... that at a minimum $\map {f''} a$ is positive. Alternatively at a maximum $\map {f'} x$ is a decreasing function of $x$ and at a minimum $\map {f'} x$ is an increasing function of $x$. Thus it is possible to discriminate quite easily between maxima and minima.*

## Source work progress

- 1963: Charles Fox:
*An Introduction to the Calculus of Variations*(2nd ed.) ... (previous) ... (next): Chapter $\text I$. The First Variation: $1.2$. Ordinary maximum and minimum theory