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

Subject Matter

 * Calculus of Variations

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 $\displaystyle I = \int_{t_0}^{t_1} \map F {q_1, q_2, \dotsc, q_n; \dot q_1, \dot q_2, \dotsc, \dot q_n; t} \rd t$
 * 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 $\displaystyle \int_{t_1}^{t_2} \map G {x, y, \dot x, \dot y} \rd t$
 * 9.10 The Weierstrassian $E$ function for $\displaystyle \int_A^B \map G {x, y, \dot x, \dot y} \rd t$
 * 9.11 Alternative forms for the $E$ function
 * 9.12 Conditions for maxima and minima of $I = \displaystyle \int_A^B \map G {x, y, \dot x, \dot y} \rd t$
 * 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.