Calculus of Variations

Calculus of Variations

This wikibook is a transcribed version of Lectures on the Calculus of Variations (the Weierstrassian theory) by Harris Hancock in 1904. The scanned original is available here from Cornell University.

CONTENTS

PREFACE

CHAPTER I: PRESENTATION OF THE PRINCIPAL PROBLEMS OF THE CALCULUS OF VARIATIONS.

• 1 The connection between the Calculus of Variations and the Theory of Maxima and Minima.

Problem I. The curve which generates a minimal surface area when rotated about a given axis.

• 2,3,4 The solution of this problem by the methods of Maxima and Minima.
• 5,6 The difference between the Calculus of Variations and the Theory of Maxima and Minima.
• 7 The coordinates $x$, $y$ expressed as functions of a parameter $t$. Problem I formulated in terms of the parameter $t$.
• 8,9 Problem II. The brachistochrone.
• 10 Problem III. The shortest line on a given surface.
• 11 The advantage of formulating the problems in terms of the parameter $t$.
• 12 Problem IV. The surface of revolution of least resistance.
• 13 The general problem stated.
• 14 The variation of the endpoints.
• 15 Problem V. The isoperimetrical problem.
• 16 Problem VI. The curve whose center of gravity lies lowest.
• 17 Statement of the general problem in Relative Maxima and Minima.
• 18 Generalizations that may be made.
• 19 Variation of a curve. Analytical definition of maximum and minimum. Neighboring curves.
• 20 A different statement of the general problem.
• 21 Inadmissibility of the presupposed existence of a maximum or a minimum.
• Problems

CHAPTER II: EXAMPLES OF SPECIAL VARIATIONS OF CURVES. APPLICATIONS TO THE CATENARY.

• 22 Total variation in the case of Problem I, Chapter I.
• 23 A bundle of neighboring curves.
• 24 The first variation.
• 25 The integral $I=\int _{x_{0}}^{x_{1}}F\left(x,y,{\frac {{\text{d}}y}{{\text{d}}x}}\right){\text{d}}x$
• 26 The vanishing of the first variation.
• 27 Application to Problem I.
• 28 The differential equation of this problem.
• 29 The integral $I=\int _{x_{0}}^{x_{1}}F(y,y'){\text{d}}x$
• 30 Solution of the differential equation of Art. 26.
• 31 The notion of a region within which two neighboring curves do not intersect.
• 32 The catenary.

CHAPTER III: PROPERTIES OF THE CATENARY.

• 33 Preliminary remarks.
• 34 The general equation of the catenary.
• 35 Geometrical construction of its tangent.
• 36 Geometrical construction of the catenary.
• 37 The catenary uniquely determined when a point on it and the direction of the tangent at this point are given.
• 38 Limits within which the catenary must lie.
• 39,40,41 The number of catenaries that may be drawn through two fixed points.
• 42 The functions $f_{1}(m)$ and $f_{2}(m)$.
• 43 The discussion of the function $f_{1}(m)$.
• 44 The discussion of the function $f_{2}(m)$.
• 45 An approximate geometrical construction for the root of a transcendental equation.
• 46,47,48 Graphical representation of the functions $f_{1}(m)$ and $f_{2}(m)$.
• 49,50 The different cases that arise and the corresponding number of catenaries.
• 50,51,52 The position of the intersection of the tangents through the two fixed points for each case.
• 53,54 The common tangents to two catenaries.
• 55 Catenaries having the same parameter which intersect in only one point.
• 56 Lindelöf's Theorem.
• 57 A second proof of the same theorem.
• 58,59 Discussion of the several cases for the possibility of a minimal surface of rotation.
• 60,61 Application to soap bubbles.

CHAPTER IV: PROPERTIES OF THE FUNCTION $F(x,y,x',y')$.

• 62 The function $F$ defined as a function of its arguments.
• 63,64,65,66,67 Necessary conditions and sufficient conditions.
• 68 The function $F(x,y,x',y')$ must be homogeneous of the first degree in $x'$ and $y'$.
• 69 Integrability of the function $F(x,y,x',y')$.
• 70 The integral $I=\int _{x_{0}}^{x_{1}}F\left(x,y,1,{\frac {{\text{d}}y}{{\text{d}}x}}\right){\text{d}}x$, when $x$ and $y$ are one-valued functions of each other.
• 71 Introduction of the variable $t$ or $s$.
• 72 Analytical condition for the function $F$.
• 73 Introduction of the function $F_{1}$.

CHAPTER V: THE VARIATION OF CURVES EXPRESSED ANALYTICALLY. THE FIRST VARIATION.

• 74 General forms of the variations hitherto employed.
• 75 The functions $\xi$ and $\eta$. Their continuity.
• 76 Neighboring curves. The first variation.
• 77 The functions $G$, $G_{1}$ and $G_{2}$.
• 78 Proof of an important lemma.
• 79 The vanishing of the first variation and the differential equation $G=0$.
• 80 The curvature expressed in terms of $F$ and $F_{1}$.
• 81 The components $w_{N}$ and $w_{T}$ in the directions of the normal and the tangent.
• 82 Variations in the direction of the tangent and in the direction of the normal.
• 83 Discontinuities in the path of integration. Irregular curves.
• 84,85 Problem of Euler illustrating the preceding article.
• 86 Summary.

CHAPTER VI: THE FORM OF THE SOLUTIONS OF THE DIFFERENTIAL EQUATION $G=0$.

• 87 Another form of the differential equation $G=0$.
• 88 Another form of the function $F_{1}$.
• 89 Integration in power-series.
• 90 The solutions $x=\phi (t,\alpha ,\beta )$, $y=\psi (t,\alpha ,\beta )$ of the differential equation $G=0$.
• 91,92 The case when $F_{1}=0$ at the initial-point of the curve.
• 93 The form of the differential equation $G=0$ when $s$ is the independent variable.
• 94 The solution of this equation.
• 95 The curve can have no singular points within the interval in question. coordinates of any point of the curve expressed as power-series in $s$.

CHAPTER VII: REMOVAL OF CERTAIN LIMITATIONS THAT HAVE BEEN MADE. INTEGRATION OF THE DIFFERENTIAL EQUATION $G=0$ FOR THE PROBLEMS OF CHAPTER I.

• 96 Instead of a sing-le regular trace, the curve may consist of a finite number of such traces.
• 97 The first derivatives of $F$ with respect to $x'$ and $y'$ vary in a continuous manner for the curve $G=0$, even if there are sudden changes in the direction of this curve.
• 98 Explanation of the result given in the preceding article.
• 99 Summary.
• 100 Solution of the differential equation for Problem I of Chapter I.
• 101,102 The discontinuous solution.
• 103 The equation $G=0$ solved for Problem II, Article 9.
• 104 The two fixed points must lie on the same loop of the cycloid.
• 105 Through two points may be drawn one and only one cycloidal-loop, which does not include a cusp.
• 106 Problem III. Problem of the shortest line on a surface.
• 107 The same result derived in a different manner.
• 108 Problem IV. Surface of rotation which offers the least resistance.
• 109,110 Solution of the equation $G=0$ for Problem IV of Chapter I.

CHAPTER VIII: THE SECOND VARIATION; ITS SIGN DETERMINED BY THAT OF THE FUNCTION $F_{1}$.

• 111 Nature and existence of the substitutions introduced.
• 112 The total variation.
• 113,114 The second variation of the function $F$.
• 115 The second variation of the integral $I$. The sign of the second variation in the determination of maximum or minimum values.
• 116 Discontinuities.
• 117 The sign of the second variation made to depend upon that of $F_{1}$.
• 118 The admissibility of a transformation that has been made. The differential equation $J=0$.
• 119 A simple form of the second variation.
• 120 A general property of a linear differential equation of the second order.
• 121 The second variation and the function $F_{1}$. The function $F{1}$ cannot change sign and must be different from $0$ and $\infty$ in order that there may be a maximum or a minimum.

CHAPTER IX: CONJUGATE POINTS.

• 122 The second variation of the differential equation $J=0$.
• 123,124 The solutions of the equations $G=0$ and $J=0$. The second variation derived from the first variation.
• 125 Variations of the constants in the solutions of $G=0$.
• 126 The solutions $u_{1}$ and $u_{2}$ of the differential equation $J=0$.
• 127 These solutions are independent of each other.
• 128 The function $\Theta (t,t')$. Conjugate points.
• 129 The relative position of conjugate points on a curve.
• 130 Graphical representation of the ratio ${\frac {u_{1}}{u_{2}}}$.
• 131 Summary.
• 132 Points of intersection of the curves $G=0$ and $\delta G=0$.
• 133 The second variation when two conjugate points are the limits of integration, and when a pair of conjugate points are situated between these limits.

CHAPTER X: THE CRITERIA THAT HAVE BEEN DERIVED UNDER THE ASSUMPTION OF CERTAIN SPECIAL VARIATIONS ARE ALSO SUFFICIENT FOR THE ESTABLISHMENT OF THE FORMULAE HITHERTO EMPLOYED.

• 134 The process employed is one of progressive exclusion.
• 135 Summary of the three necessary conditions that have been derived.
• 136,137 Special variations. The total variation.
• 138 A theorem in quadratic forms.
• 139 Establishment of the conditions that have been derived from the second variation.
• 140,141,142,143,144 Application to the first four problems of Chapter I.

CHAPTER XI: THE NOTION OF A FIELD ABOUT THE CURVE WHICH OFFERS A MINIMUM OR A MAXIMUM VALUE OF THE INTEGRAL. THE GEOMETRICAL MEANING OF THE CONJUGATE POINTS.

• 145 Notion of a field.
• 146 Neighboring curves which belong to the family of curves $G=0$.
• 147 A general theorem in the reversion of series.
• 148 The coordinates of a neighboring curve expressed in power-series of $k$, where $k$ is the trigonometric tangent between the initial directions of the neighboring curve and the original curve.
• 149 A curve which satisfies the equation $G=0$ is determined as soon as its initial point and the direction of the tangent at this point are known.
• 150 Limits assigned to $k$. Extension of the notion of a field.
• 151 Intersection of two neighboring curves. Conjugate points.
• 152 A point cannot be its own conjugate. The derivative of $\Theta (t,t')$ does not vanish at a point which causes the function itself to vanish.

CHAPTER XII: A FOURTH AND FINAL CONDITION FOR THE EXISTENCE OF A MAXIMUM OR A MINIMUM, AND A PROOF THAT THE CONDITIONS WHICH HAVE BEEN GIVEN ARE SUFFICIENT.

• 153 The notion of a field continued from the preceding Chapter.
• 154 The function ${\mathcal {E}}(x,y,p,q,{\bar {p}},{\bar {q}})$.
• 155 The function ${\mathcal {E}}$ must have the same sign for every point of the curve.
• 156 The sufficiency of the above condition.
• 157 Another form of the function ${\mathcal {E}}$.
• 158 Still another form.
• 159,160 The signs of the function ${\mathcal {E}}$ and $F_{1}$.
• 161 Another proof of the sufficiency of the condition as given in Article 156.
• 162 The function ${\mathcal {E}}$ cannot be zero along an entire curve in the given field.
• 163 The envelope of conjugate points.
• 164 The curve may be composed of a finite number of regular traces.
• 165 Cases where the traces are not regular.
• 166 Generalizations in the Integral Calculus.
• 167,168,169,170,171,172 Applications to the four problems already considered.
• 173 When $F(x,y,x',y')$ is a rational function of $x'$ and $y'$, there can exist neither a maximum nor a minimum value of the integral.
• 174 General summary.
• 175 Extensions and generalizations: Instead of the determination of a structure of the first kind in the domain of two variables, it may be required to determine a structure of the first kind in the domain of $n$ quantities.
• 176 When equations of condition exist among the variables.
• 177 When the second and higher derivatives appear.
• 178 The Calculus of variations applied to the determination of structures of a higher kind. The minimal surfaces.

RELATIVE MAXIMA AND MINIMA

CHAPTER XIII: STATEMENT OF THE PROBLEM. DERIVATION OF THE NECESSARY CONDITIONS.

• 179 The general problem stated.
• 180 Existence of substitutions by which one integral remains unchanged while the other is caused to vary. An exceptional case.
• 181 Case of two variables. Convergence of the series that appear.
• 182 The nature of the substitutions that have been introduced.
• 183 Formation of certain quotients which depend only upon the nature of the curve.
• 184 Generalization, in which several integrals are to retain fixed values.
• 185 The quotient of two definite integrals being denoted by $\lambda$, it is shown that $\lambda$ has the same constant value for the whole curve.
• 186 The differential equation $G^{(0)}-\lambda G^{(1)}=0$.
• 187 Extension of the theorem of Article 97.
• 188 Discontinuities, etc.
• 189 The second variation: the three conditions formulated in Article 135 are also necessary here.

CHAPTER XIV: THE ISOPERIMETRICAL PROBLEM.

• 190 Statement of the problem.
• 191 A simpler form of the integral that appears.
• 192 The function $F_{1}$ for this problem.
• 193 Integration of the differential equation that occurs.
• 194 An immediate consequence is the theorem of Steiner: Those portions of curve that are free to vary, are the arcs of equal circles.
• 195 If there exists a curve, which with a given periphery incloses the greatest surface area, that curve is a circle.
• 196 The admissibility that this property belongs to the circle.

CHAPTER XV: RESTRICTED VARIATIONS. THE THEOREMS OF STEINER.

• 197 Variations along two different portions of curve.
• 198 Variation where a point must remain upon a fixed curve.
• 199 Application to a particular case.
• 200 Variation where a part of the curve coincides with a fixed curve.
• 201 Generalizations involving several variables and several integrals.
• 202 The isoperimetrical problem when the circle (Art. 195) which incloses the given area cannot be inscribed within fixed boundaries.
• 203 Statement of two problems due to Steiner. Criticism of his assertion that the Calculus of Variations was not sufficient for the proof of these problems.
• 204 Two problems due to Weierstrass which are more general than Steiner's and their proof by means of the Calculus of Variations.
• 205 The behavior of the ${\mathcal {E}}$-function at fixed boundaries.
• 206 Further discussion regarding this function.
• 207 The case that there is a sudden change in the direction of the boundary curve at the point where it is approached by the curve that is free to vary.
• 208 The case where the curve meets the boundary at a point and then leaves it.
• 209 The tangents to the two portions of curve make equal angles with the tangent to the fixed curve.
• 210 The isoperimetrical problem reversed.
• 211 Consideration of the problem: Three points not lying in the same straight line are given in the plane. It is required to draw a line through them in a definite order which with a given length includes the greatest possible surface area.
• 212 Expression for portions of the curve that overlap.
• 213 The solution of the differential equation may be straight lines or arcs of circles.
• 214 The problem reduced to a problem in the Theory of Maxima and Minima.
• 215 The problem solved.

CHAPTER XVI: THE DETERMINATION OF THE CURVE OF GIVEN LENGTH AND GIVEN END-POINTS, WHOSE CENTER OF GRAVITY LIES LOWEST.

• 216 Statement of the problem.
• 217 The necessary conditions
• 218 The number of catenaries having a prescribed length that may be through two given points with respect to a fixed directrix.
• 219 The constants uniquely determined.

CHAPTER XVII: THE SUFFICIENT CONDITIONS.

• 220 The problem solved without using the second variation.
• 221 The ${\mathcal {E}}$-function.
• 222 Consequences due to this function.
• 223 A field about the curve which maximizes or minimizes the integral.
• 234 Further discussion of the nature of the enveloped space.
• 225 Properties of an enveloped space which lies within the first enveloped space.
• 226 The sufficiency of the ${\mathcal {E}}$ condition.
• 227 It is assumed that through the initial-point and any other point in the enveloped space only one curve may be drawn, which satisfies the differential equation (cf. Article 230).
• 228 The ${\mathcal {E}}$-function cannot be zero along an entire curve that lies within the enveloped space (cf. Article 230).
• 229 Extension of the meaning- of the integrals that have been employed.

CHAPTER XVIII: PROOF OF TWO THEOREMS WHICH HAVE BEEN ASSUMED IN THE PREVIOUS CHAPTER.

• 230 The two theorems proved in the present Chapter are:

1) That possible to construct a portion of space about a curve which satisfies the differential equation of the problem in such a way that it is always possible to join any point in this limited space and the initial-point by one and only one curve which likewise satisfies the differential equation.

2) That the function ${\mathcal {E}}$ cannot vanish along an entire curve within such a portion of space.

• 231 The equations of the curve in space. The coordinates expressed in power-series.
• 232 The curve determined through its initial-point and initial direction. Condition that two curves pass through the same initial-point and have initial directions differing by a given small quantity.
• 233 Condition that one of these curves pass through a point which is in the neighborhood of a point on the other curve. The determinant $D(t_{0},t)$ which arises from these conditions.
• 234 Conjugate points in connection with this determinant.
• 235 Case where the end-points are conjugate to each other.
• 236 The differential equation $G=0$ and the determinant $D(t_{0},t)$.
• 237 Simplification of the determinant $D(t_{0},t)$.
• 238,239,240,241,242 The determinant $D(t_{0},t)$ changes sign when it vanishes.
• 241 An exceptional case.
• 243 The point conjugate to the initial-point is the limit of the intersection of neighboring curves that pass through the initial-point.
• 244 When in the determinant $D(t_{0},t)$, the quantity $t_{0}$ varies in a continuous manner, the point conjugate to $t_{0}$ varies in a continuous manner.
• 245,246,247,248 Proof of the theorem that a portion of curve including conjugate points may be so varied that the total variation of one of the integrals may be either positive or negative, while the other integral remains unchanged.
• 249,250,251,252 The function ${\mathcal {E}}$ cannot vanish along an entire curve within the portion of space defined above.
• 253,254,255,256 The ${\mathcal {E}}$-function for the case of the Isoperimetrical Problem.
• 256,257 The ${\mathcal {E}}$-function for the curve whose center of gravity lies lowest.