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

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

Published $\text {1963}$, Dover Publications

ISBN 0-486-65499-0

### 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
5.19 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.

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## 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.